Lore and Science in Ancient Pythagoreanism
Walter Burkert
Translated by Edwin L. Minar, Jr.
Harvard University Press
Cambridge, Massachusetts
1972
Zea ft aah Altea do os ‘AE EE Oa 1. PE PTEPOP AT?
266159
OCCIDENTAL COLLEGE LIBRARY
© Copyright 1972 by the President and Fellows of Harvard College
All rights reserved
English edition, translated with revisions from Weisheit und Wissenschaft:
Studien zu Pythagoras, Philolaos und Platon, Copyright 1962 by
Verlag Hans Carl, Niirnberg
Library of Congress Catalog Card Number 70-1628 56
SBN 674-53918~4
Printed in the United States of America
Preface to the German Edition
If Pythagoras does not present himself to our minds as a sharply
outlined figure, standing in the bright light of history, this is not merely
the result of accidents in the course of historical transmission. From the
very beginning, his influence was mainly felt in an atmosphere of
miracle, secrecy, and revelation. In that twilight period between old
and new, when Greeks, in a historically unique achievement, were
discovering the rational interpretation of the world and quantitative
natural science, Pythagoras represents not the origin of the new, but the
survival or revival of ancient, pre-scientific lore, based on super-
human authority and expressed in ritual obligation. The lore of
number is multifarious and changeable. That which was later regarded
as the philosophy of Pythagoras had its roots in the school of Plato.
Outlines of an earlier reformulation of Pythagorean doctrine in the
manner of the φυσιολογία of the fifth century can be detected in the
fragments of Philolaus. As the old and the new interpenetrated and
influenced each other, the picture of Pythagoras became distorted until,
with the victory of rational science, he came to seem its true founder.
To investigate these interrelationships is still a somewhat risky
undertaking; but an attempt has been made to take more account than
has previously been done of the variety in the kinds of evidence
available and, above all, to clarify the ramifications and the divagations
of the tradition.
I am indebted for much advice and encouragement to my teachers
Otto Seel, Reinhold Merkelbach, and Helmut Berve, as well as to
Ir. Ludwig Koenen and to Dr. Burkhart Cardauns, who also helped
me with the proofs. To all of these I offer hearty thanks.
Erlangen Walter Burkert
April 1962
Preface to the English Edition
In revising this book for translation, it has been impossible to add
references to all the literature on the subject which has appeared since
1962. I have tried to concentrate on the ancient evidence, to cut down
polemics, and to incorporate whatever I have learned in these years,
notably from some reviews of the German edition, and from con-
tinuous discussions with B. L. van der Waerden. In the question of the
“discovery of the irrational,” I have taken a stand which is less critical
of the tradition; and more thorough acquaintance with ancient religion
has pushed the concept of “‘shamanism”’ further into the background.
But though a good number of passages have been revised, and though
there are some small rearrangements in the order of treatment, still the
book has remained, in all the main lines, the same.
New and comprehensive accounts of Pythagoreanism have been
given by Kurt von Fritz, H. Darrie, and B. L. van der Waerden in
Pauly-Wissowa (XXIV 171-300; Supp. X 843-864), and by W. K. C.
Guthrie in his History of Greek Philosophy, vol. I; and the same year
(1966) saw the appearance of two books with the title Pythagoras and
Early Pythagoreanism, by C. J. de Vogel and J. A. Philip. Discussion is
certain to continue. I cannot claim to have provided a definitive
history of Pythagoreanism, or a complete account of Pythagoras, the
man and the genius. Still I trust that this book gives a full and perspic-
uous presentation of the evidence and thus will be useful even to those
who are not inclined to draw the same conclusions from it.
My special thanks are due to Edwin L. Minar, Jr., who not only
completed the laborious task of translation in a spirit of most pleasant
collaboration, but to whom is due the original initiative which brought
about the English edition. The responsibility for the content, and for
all that may be wrong in it, remains mine.
Schwerzenbach-Ztirich Walter Burkert
August 1970
Contents
Introduction
. Platonic and Pythagorean Number Theory
. The Philosophy of the Pythagoreans according to Aristotle
. The Later Non-Aristotelian Tradition and Its Sources,
Speusippus, Xenocrates, and Heraclides Ponticus
. Pythagoreanism in Plato and the Origin in Platonism of the
Pythagorean Tradition
I
- 1, The Platonic Theory of Ideal Numbers
2
3
II. Pythagoras in the Earliest Tradition
A Ὁ. Ww HN &
. Source Problems
. Historical Background
. Metempsychosis and “Shamanism”
. Acusmata
. Acusmatici and Mathematici
. Early Evidence for Pythagoras as a Scientist?
UI. Philolaus
be
3
|
]
2
3
4
. The Special Position of Philolaus’ Book in the Pythagorean
Tradition
. The Spurious and the Genuine in the Philolaus Fragments
. Reflections of Pythagorean Philosophy in the Fifth
Century B.c.?
V. Astronomy and Pythagoreanism
. The Structure of the World and the Planetary System
. The Theory of Planetary Movements
. The Cosmos of Philolaus
. Harmony of the Spheres and Astral Immortality
Contents
V. Pythagorean Musical Theory 369
1. Speculation, Experimentation, and Fiction a 369
2. Number Symbolism and Calculation of Proportions in 386
Philolaus
VI. Pythagorean Number Theory and Greek Mathematics 401
1. Did the Pythagoreans Lay the Foundations of Greek Mathe- 401
matics?
2. Pythagorean Arithmetic 427
3. Pythagorean Geometry and Mathematical Secrets 447
4. Number and Cosmos 465
Abbreviations 485
Bibliography . 493
Greek Words Discussed $09
Index of Passages 510
General Index $28
Introduction
The “Pythagorean question” has sometimes been compared with
the Homeric question. Not that the details of the problem would
especially suggest this; what does remind one of that most famous of
philological controversies is the difficulty of the argument and the lack
of agreement on methodology, as well as the multiplicity and contra-
dictory character of the solutions advanced. Another similarity, and
not the least striking, lies in the tremendous importance of the questions
about the life, activity, and influence of Pythagoras of Samos. Over
the origins of Greek philosophy and science, as over the beginning of
Greek literature, lies the shadow of a great traditional name. The
attempts of scholarship to grasp the underlying historical reality keep
getting entangled in contradictions; where some think they discern the
figure of a world-historical genius, others find little more than empty
nothingness. |
Pythagoras’ influence was a lasting one. The ancient tradition of the |
history of philosophy made him the ancestor of the “Italian School”
and therefore, after Thales, the second, and more important, originator
of philosophia—in fact, the inventor of the word. The doctrine trans-
mitted under his name, that numbers are the principles of what exists, |
that the “One” is its primal ground, beeame-part.of the-amales
Neoplatonism:—n-the-tremdser-by-tambtichus, Pythagoras was the
high priest, par excellence, of the divine wisdom. He-then_became;-in
the trivializing school tradition of the Middle Ages, the master of the
quadrivium, and in particular the inventor of arithmetic. The early
modern period discovered Pythagoras as the creator of natural science,
which was just then being reborn; what Copernicus and Galileo
(aught was regarded by their contemporaries as a revival of Pythag-
orcan science.
Subsequently, as research based on historical perspective increasingly
'Hg., Campanella in a letter to Galileo of Jan. 13, 1611 (T. Campanella, Lettere, ed.
V. Spampanato | Bari, 1927] 164). Further literature in Capparelli, [ 20; see below,
ch. TV 3, nn. t-a.
INTRODUCTION
replaced the uncritical appropriation of ancient culture, the traditional
picture of Pythagoras, imposing though also vague in outline, in-
evitably gave way more and more before criticism. In the scholarly
controversy that followed scarcely a single fact remained undisputed,
save that in Plato’s day and then later, in the first century B.c., there
were Pythagoreioi. The “wisdom of Pythagoras,” however, has also
had passionate defenders, who opposed to criticism a countercritique:
one-sided, self-sufficient methodology, they protested, had substituted
hypothesis for tradition.
When we set out to survey the most important attitudes and trends
in modern Pythagorean scholarship, the point of departure must be
the work of Eduard Zeller.? In it the material is not only collected, with
a completeness scarcely to be surpassed, but sifted with uncommon
methodological rigor. The criterion for the value of a tradition is its
age, and Zeller arrives at the verdict, often cited since in agreement and
disagreement: ‘‘The tradition about Pythagoreanism and its founder
thus has more and more to tell, the further it lies, chronologically,
from the events...” (I 364). This “expansion of the tradition ” (ibid.)
arose basically from “dogmatic preconceptions, partisan interests,
dubious legends, and spurious writings’’ (365). The most important
source, nearly the only one which is left, is in the reports of Aristotle,
in his surviving treatises. A second primary source is found in the
fragments of Philolaus, in which August Boeckh, in his day, claimed
to have found a firm foothold amidst the bog of Pythagorean pseud-
cpigrapha.? But Aristotle speaks of “‘Pythagoreans,” not of “Pythag-
oras’’; so the figure of the master fades off in a mist of nonhistorical
legend. Pythagoras is still recognized as the “founder of a religious
socicty” and teacher of transmigration (411), but all philosophical
significance is denied his ethical and religious doctrines (557ff). Along-
side this stands, without connection, the number philosophy of the
Pythagorcans. This can be reconstructed from Aristotle and is confirmed
by Philolaus, though aside from him the Pythagoreans remain anony-
mous and scarcely datable. Sharply separated from all this ancient
material is the neo-Pythagorcan school, which (Zeller thinks) arose,
compounded of Platonic and Aristotelian clements, not before the
first century B.C. (III 2.92ff).
*1 461-619, ΠῚ 2.92-254. The text is essentially that of the second edition (1856).
Zeller summarized his position under the tide Pythagoras und die Pythagoraslegende,”
Vortr, ας Abb. 1 (Leipzig, 1864) 30-50.
* Hut contrary to Boeckh, Zeller pronounced the fragment on the world soul (21)
spurious (1476.1); cf below, ch. ΠῚ 2, mn. 20.
Introduction
Zeller’s ‘work had a decisive influence, especially in Germany; it
dominates Dicls’s arrangement of testimonia in the Fragmente der
Vorsokratiker. In the chapter on Pythagoras (DK 14), the biographical
reports are assembled, along with the reports as to whether or not
Pythagoras wrote anything. In the separate chapter on “The
Pythagorean School” (DK 58), the most important section is the
collection of “ancient Peripatetic” material (58B),5 following Iambli-
chus’ catalogue of Pythagoreans (58), and itself followed by the
acusmata (58), the Pythagorikai apophaseis of Aristoxenus (58D), and
the allusions to ““Pythagorists”’ in the Middle Comedy (58E). Between
these two chapters are found, in chronological order as far as possible,
the testimonies on individual Pythagoreans, notably Hippasus (DK 18),
Philolaus (DK 44), and Archytas (DK 47). Hermann Diels follows
Zcller also in the question of the genuineness of the Philolaus fragments.
Zeller’s solution, however, left a number of problems unsolved, and
later research entered in with supplement, modification, and criticism.
Above all, a gap had opened between Pythagoras the religious founder
and the number philosophy of anonymous Pythagoreans; to connect
these disparate elements and to show their original unity was bound
to be an extremely enticing challenge. For at thistime the tide of system
building in philosophy, which had borne the work of Zeller, was
ebbing, and this very change made possible a deeper understanding of
cultural history. As the boundaries of philosophy became fluid, the
connections with pre-scientific, religious-mythical thought became
clearer. So the task appeared to be to comprehend how religion and
philosophy could be united in Pythagoras: mysticism and science (or
at least the germ from which science sprang). Then came the high
tide of the evolutionary idea, and it began to seem possible, with its
help, to explain the contradictions of the tradition and to give every-
thing its place in an extended and detailed history of Pythagoreanism.
The direction was set by August Déring (1892); his thesis became
most influential because he was followed by John Burnet in the later
editions of his Early Greek Philosophy.’ The unity of science and religion
is found in the ideal of catharsis; scientific activity is the highest form
‘Two doxographical reports are arbitrarily added (14.20, 21; cf. below, ch. IV 1,
11. 39-41; 1 3, n. 1§ 1-153).
\ rom the fifth edition on, the excerpt from the Hypomnemata has been added, as a
result of the discussion of Max Wellmann (s8B1a; cf. below, ch. 13, ἢ. 1).
"Surnet refers to Déring (EGP 98 αν 3); as a result, the extent of what Burnet traced
back to Pythagoras himself increased considerably after the first edition (London, 1892,
1.0.4}.
INTRODUCTION
replaced the uncritical appropriation of ancient culture, the traditional
picture of Pythagoras, imposing though also vague in outline, in-
evitably gave way more and morc before criticism. In the scholarly
controversy that followed scarcely a single fact remained undisputed,
save that in Plato’s day and then later, in the first century B.c., there
were Pythagoreioi. The “wisdom of Pythagoras,” however, has also
had passionate defenders, who opposed to criticism a countercritique:
one-sided, self-sufficient methodology, they protested, had substituted
hypothesis for tradition.
When we set out to survey the most important attitudes and trends
in modern Pythagorean scholarship, the point of departure must be
the work of Eduard Zeller.? In it the material is not only collected, with
a completeness scarcely to be surpassed, but sifted with uncommon
methodological rigor. The criterion for the value of a tradition is its
age, and Zeller arrives at the verdict, often cited since in agreement and
disagreement: “The tradition about Pythagoreanism and its founder
thus has more and more to tell, the further it lies, chronologically,
from the events...” (I 364). This “expansion of the tradition” (ibid.)
arose basically from “dogmatic preconceptions, partisan interests,
dubious legends, and spurious writings” (365). The most important
source, nearly the only one which is left, is in the reports of Aristotle,
in his surviving treatises. A second primary source is found in the
fragments of Philolaus, in which August Boeckh, in his day, claimed
to have found a firm foothold amidst the bog of Pythagorean pseud-
epigrapha.’ But Aristotle speaks of ‘“Pythagoreans,’ not of ‘Pythag-
oras’’; so the figure of the master fades off in a mist of nonhistorical
legend. Pythagoras is still recognized as the “founder of a religious
socicty”” and teacher of transmigration (411), but all philosophical
significance is denied his ethical and religious doctrines (557ff). Along-
side this stands, without connection, the number philosophy of the
Pythagoreans. This can be reconstructed from Aristotle and is confirmed
by Philolaus, though aside from him the Pythagoreans remain anony-
mous and scarcely datable. Sharply separated from all this ancient
material is the nco-Pythagorean school, which (Zeller thinks) arose,
compounded of Platonic and Aristotelian clements, not before the
first century B.C. (III 2.92ff).
21 361-617, ΠῚ 2.92-254. The text is essentially that of the second edition (1856).
Zeller summarized his position under the title “Pythagoras und die Pythagoraslegende,”
ort, uw. ἄς. 1 (Leipzig, 1865) 30-$0.
“But contrary to Boeckh, Zeller pronounced the fragment on the world soul (21)
spurious (1 476.1); cf below, ch. M2, a. 20.
Introduction
Zeller’s ‘work had a decisive influence, especially in Germany; it
dominates Dicls’s arrangement of testimonia in the Fragmente der
Vorsokratiker. In the chapter on Pythagoras (DK 14), the biographical
reports are assembled, along with the reports as to whether or not
Pythagoras wrote anything. In the separate chapter on “The
Pythagorean School” (DK 58), the most important section is the
collection of “ancient Peripatetic’’ material (58B),° following Iambli-
chus’ catalogue of Pythagoreans (58A), and itself followed by the
acusmata (58C), the Pythagorikai apophaseis of Aristoxenus (58D), and
the allusions to “Pythagorists”’ in the Middle Comedy (58E). Between
these two chapters are found, in chronological order as far as possible,
the testimonies on individual Pythagoreans, notably Hippasus (DK 18),
Philolaus (DK 44), and Archytas (DK 47). Hermann Diels follows
Zeller also in the question of the genuineness of the Philolaus fragments.
Zeller’s solution, however, left a number of problems unsolved, and
later research entered in with supplement, modification, and criticism.
Above all, a gap had opened between Pythagoras the religious founder
and the number philosophy of anonymous Pythagoreans; to connect
these disparate elements and to show their original unity was bound
to be an extremely enticing challenge. For at thistime the tide of system
building in philosophy, which had borne the work of Zeller, was
cbbing, and this very change made possible a deeper understanding of
cultural history. As the boundaries of philosophy became fluid, the
connections with pre-scientific, religious-mythical thought became
clearer. So the task appeared to be to comprehend how religion and
philosophy could be united in Pythagoras: mysticism and science (or
at least the germ from which science sprang). Then came the high
tide of the evolutionary idea, and it began to seem possible, with its
help, to explain the contradictions of the tradition and to give every-
thing its place in an extended and detailed history of Pythagoreanism.
The direction was set by August Déring (1892); his thesis became
most influential because he was followed by John Burnet in the later
cditions of his Early Greek Philosophy.6 The unity of science and religion
is found in the ideal of catharsis; scientific activity is the highest form
4‘wo doxographical reports are arbitrarily added (14.20, 21; cf. below, ch. IV 1,
rite. 39-413 [ 3, ἢ. 1§1-153).
"Brom the fitth edition on, the excerpt from the Hypomnemata has been added, as a
result of the discussion of Max Wellniann (s8B1a; cf below, ch. I 3, n. 1).
“Burnet refers to Déring (EGP δ αν 3); as a result, the extent of what Burnet traced
hack to Pythagoras himself inereased considerably after the first edition (London, 1892,
(0.46.
INTRODUCTION
of purification and conduces toward the salvation of the soul.’ Further,
we have the principle that “The more primitive any Pythagorean
doctrine appears, the more likely it is to be that of Pythagoras himself”’
(Burnet, EGP 99); in this way it is inferred that Pythagoras had not only
a number theory but an astronomical system. A more advanced stage
is discernible in the “Pythagoreans’’ of Aristotle, but his reports too
are broken down into different categories and are supplemented from
Plato as well as later sources.
The picture of Pythagoreanism built up in this manner is far more
colorful than Zeller’s skeptical account, but it depended so largely
on inferences and hypotheses that opposition was inevitable. On the
basis of the same leading idea, of development from religion to rational
science, completely different and mutually irreconcilable reconstruc-
tions were offered. The most significant contribution, after Burnet,
was F. M. Cornford’s article “ Mysticism and Science in the Pythago-
rean Tradition” ;® but his conclusions were importantly modified by
his own pupil J. E. Raven.® Here should be mentioned also the in-
dependent accounts of Léon Robin, Abel Rey, and Pierre-Maxime
Schuhl. The fact that each scholar had to erect a whole new super-
structure shows the weakness of the foundation.
Nevertheless, the attempts to improve on Zeller are based on addition-
al source material, which, though not newly discovered, was for the
first time revealed in its true purport. For this was a further weakness
in Zeller’s position; he had been too hasty in rejecting the later tradition.
More exact study showed, here as elsewhere, the fallacy in the principle
recentiores, ergo deteriores. Erwin Rohde initiated the careful analysis of
the sources of the lives of Pythagoras by Porphyry and Iamblichus,
and this led back, for substantial portions, to authors of the fourth
century B.c.—Aristoxenus, Dicaearchus, Heraclides Ponticus, then
Timaeus. With growing optimism others followed along the path
Rohde had pioneered.!° It was not only in the realm of political
history that many new insights could thus be attained; the problem of
the philosophy and science of Pythagoras also began to appear in a
? Déring, AGP 1892, 505; Burnet, EGP 97f; below, ch. II 6.
8 CQ 1922-1923; cf. PlParm 1-27 and CAH IV (1926) 544-552.
® Raven PyElI, repeated in more cautious tones in KR 216ff, 307ff.
10 See below, ch. II 1. Special gratitude is due Armand Delatte, who completely studied
through the later tradition, including even its most abstruse areas. All the parallel material
is collected and set out very perspicuously by Delatte in his edition of Diogenes Laertius’
life of Pythagoras, and by Ludwig Deubner, building upon Delatte’s work, in_ his
edition of Iamblichus’ life of Pythagoras.
Introduction
new light. The reconstruction of Aristotle’s Protrepticus from passages
in Iamblichus seemed to yield statements by Aristotle himself about
the philosopher Pythagoras." But most important of all, thirty-five
years after the first appearance of Zeller’s Philosophie der Griechen,
Diels had shown in his Doxographi graeci (1879) that Theophrastus
was the ultimate source of the wide-branching doxographical tradition.
After that it was difficult to attribute the reports of the Placita to the
neo-Pythagoreans, as Zeller had done (I 467ff). Diels classified them
with the “ancient Peripatetic tradition” (DK 58B15), and they play
their important role in all the above-mentioned reconstructions of a
developing Pythagoreanism.
There was also an attempt to discover indirect sources; one sought
to find Pythagorean material reflected in other pre-Socratic thinkers,
whether by way of influence or of polemic. The most important step
in this direction was the thesis of Paul Tannery, that Parmenides took
the cosmology of the doxa section of his poem from the Pythagoreans
and that Zeno’s polemic was directed against their number theory.
‘This brought some very ancient evidence into the field, which could
help to classify, to supplement, and even to test the reports of Aristotle.
‘This hypothesis was developed in many ways, and only very seldom
placed seriously in question? Similar endeavors were of course
directed toward Plato, whose connection with Archytas is firmly
fixed in the biographical tradition. To be sure, it is especially difficult
in this case to achieve any certainty. The native capacity of Plato’s
imind to remold and reshape is too great, and in each specific instance
i ix a controversial question to what extent particular statements of
the dialogues may be interpreted as historical facts. The theory of
Burnet and Taylor, which takes every portrayal in the dialogues as
historical fact, not only transforms the entire Timaeus to a Pythagorean
document of the fifth century B.c., but presents us with a Socrates who
ts an advocate of the theory of ideas and an adept of Pythagorean
wisdom. Though this radical solution has attracted no following,
the Pythagorean origin of the theory of ideas, and especially of the
"hat. Protr. 61.78; Arist. Protr. fr, 11 Walzer. On this, cf. Burkert, Hermes 1960,
intl, below, ch. 13, 0.77. 1. Diiring, Aristotle’s Protrepticus (Goteborg, 1961) p. 189, holds
tothe Aristotelian origin of this passage in Iamblichus (B18-20 Diiring); this forces him
toamooth over the difficule transition at lam. Profr. p. $1.6f by conjecture and to ignore
the clusaecteristic τίς dori, at lam. Profr. 4.11; asa result, the hints toward source analysis
that the text of lamblichus provides are arbitrarily removed.
Mt, below, ch. HD 3 and VI 3, n. 46.
“6 Burnet, GP 2776, jos; {ΠΡΕ agit; Taylor, Vin. and VarSecr (Oxford, tort)
LIE
INTRODUCTION
doctrine of recollection, is still being discussed; and it is gencrally
taken as proved that Plato owed his scientific knowledge to the Pythag-
orcans, especially in the realm of astronomy.
Zeller had scarcely more than touched on the natural sciences, never
doubting the fundamental importance of the Pythagoreans in the
development of Greek mathematics, astronomy, and music (I 4osf).
Here was a further incentive to look for more precise results. After the
fundamental investigations of Paul Tannery, the comprehensive
expositions of Sir Thomas Heath, and the competent studies of Abel
Rey, detailed reconstructions of Pythagorean mathematics were given
by Oskar Becker, B. L. van der Waerden, and Kurt von Fritz. Even
though every attribution to Pythagoras himself remains unprovable,
still the reconstruction seems to lead back almost as far as his era.
Thus analyses and reconstructions have in many ways given occasion
to think less skeptically than Zeller about the sources for Pythagorean
history. More and more the tendency has been to put a higher valuation
on the later tradition. Even apocryphal works were, correctly, taken
seriously as evidence pertinent to intellectual history, and some were
credited with an early date, in particular the Memoirs (Hypomnemata)
that had been excerpted by Alexander Polyhistor.!? Research in the
religious history of late antiquity led in a similar direction. Franz
Cumont and Jéréme Carcopino made extensive use of Pythagorean
tradition in the interpretation of funerary symbolism of the imperial
period. From this point of view there was no difference discernible
between early and late Pythagoreanism; it was rather as though a
powerful and continuous stream flowed from an ancient source.
The numerous studies of Pierre Boyancé also follow this tendency ;
their aim is to grasp the “origine pythagoricienne” behind late
material. |
Understandably, the “Italian philosophy” was interpreted (and still
is) in an even more trustful spirit in Italy. Augusto Rostagni’s book
I! verbo di Pitagora may be regarded as the most ingenious, though at the
same time the most daring, attempt to comprehend science and mysti-
cism as a unity and to trace them back to Pythagoras himself.1® The
concern of Rodolfo Mondolfo has been cautious but resolute defense
CF. below, ch. 1.2, n. 82: II 6, nn. 22-24.
'® On Plato's astronomy, below, ch. IV 1-2; cf. also H. Cherniss, Lustrum 4 (1959) Soff.
"Cf below, ch. VI.
"Chon 5. Delatte, Pol. r2atff maintains that Archytas’ On Law and Justice is genuine
(but see below, ch. 13, n. 1§6). See also Mondolfo in ZAM 252}
"On this, cf below, ch. ΠΕ 1, ἡ. 7: see also ch. Ht, 47; Hea, πν 18.
Introduction
of the tradition against the attacks of critics.!9 Even the translations
and explications of the Pythagorean source material collected in the
Fragmente der Vorsokratiker, by Antonio Maddalena (1954) and Maria
Timpanaro Cardini (1958-1964), have the goal of confirming the age
and philosophical significance of Pythagoreanism; and chauvinistic
enthusiasm for Pythagoras runs riot in the bulky works of Vincenzo
Capparelli.?°
Over against all these attempts to achieve a more positive view than
Zeller are energetic movements of scholarly criticism which have even
called into question testimony accepted by that scholar. The genuine-~
ness of the Philolaus fragments was attacked by Carl Schaarschmidt
in 1864, and in 1868 by Ingram Bywater. While Zeller’s authority
held up for a while in Germany, Burnet followed Bywater and there-
with ensured the predominance of the negative verdict on the Philolaus
fragments which still holds in the English-speaking world.2!
Rejection of these Philolaus fragments is an essential element in
the thesis of Erich Frank, whose book Plato und die sogenannten Pythag-
oreer (1923) towers over everything else that has appeared since Zeller
on the history of Pythagoreanism, in the qualities of critical vigor,
penetration, and firmness of judgment. To be sure, its merits are
counterbalanced by one-sidedness and obvious perversities. Frank’s
methodological contribution was that he consistently held to the history
of the natural sciences—mathematics, music, and astronomy—as
basis for the reconstruction of Pythagoreanism. The first result was
to date the development much later: all Pythagorean science, he
thought, had come into existence in the circle of Archytas, about
400 B.C., influenced by the fully developed atomism of Democritus.
The philosophy of the “so-called Pythagoreans,” however, the
number theory, was dependent on the late Plato, and was basically
a creation of Speusippus, who had also himself forged the book
attributed to Philolaus.22 The figure of Pythagoras fades into the mist,
"Ὁ Mondolfo added to his translation of Zeller an ample “ Nota sulle fonte della cono~
scenZa 6 ricostruzione storica del Pitagorismo”’ (313-38 5), and published two parts of it
separately: ‘‘Sui frammenti di Filolao,” RivFil 15 (1937) 225-245 (=ZM 367-381), and
‘Platone ς la storia del Pitagorismo,” AGR 39 (1937) 235-251 (=ZM 335-345).
“0 La sapienza di Pitagora (Padua, 1941-1944), 2 vol. with 1,536 pages; I] contributo
pitagorico alla scienza (Padua, 195 5); and I! tenore di vita pitagorico ed il problema della
omotosis (Padua, 1958).
*l CE Zeller [| 369.3 (vs. Schaarschmidt); below, ch. ΠῚ I, ἢ. 14.
“This thesis was advanced by Ernst Flowald, too, in Essays on the History of Medicine
Presented to Karl Sudhoff (Ziivich, 1924, pp. 63-72; see his very favorable discussion of
rank’s book, JAW 197 [1929] 1004}, and worked out, though in a somewhat superficial
Manner, in the dissertation af Jenny Hollinger (Ztirich, 1Y25),
INTRODUCTION
and all the much-discussed Pythagorcanism of the fifth centur
becomes a mirage. /
Frank’s book, teeming with arbitrary theories and ex parte jude-
ments, has been severely criticized ;23 and curiously enough, Frank
himself expressed quite different views in later writings.®4 Nevertheles
the book still has lnportance, above all because of the extreme way
which the problem is put: “Plato and the Pythagoreans”’—their
mutual relationship is in fact the central problem of any historical
investigation of Pythagoreanism, and Frank was right in perceiving
that the influence did not go entirely in one direction. There is Platonic
material which at a later date was wrongly labeled Pythagorean, and
the generation of Plato’s immediate disciples—Speusippus, ore
crates, and Heraclides—played the decisive role in this development.
Just as correct, and important, is the idea of regarding “Pythagorean”
mathematics and science in the context of non-Pythagorean science
which stems from the Ionian and Eleatic philosophy of nature, iad
inquiring to what extent the Pythagorcans may have been on the
receiving side.
In fact, notwithstanding the reconstructions of “ Pythagorean
mathematics,” the date and importance of Pythagorean influence on
Greek mathematics had been called into question as long ago as 1907 by
Gustav Junge, then by Heinrich Vogt, and above all by the profound
and influential work of Eva Sachs, Die fiinf platonischen Kérper ( 1917).
The reports relating to Pythagoras himself are eliminated, what is
firmly attested is dated later, and the non-Pythagorean science of the
Greeks is revealed. Finally, William Arthur Heidel tried to push further
on in this direction.” If we add that the reports of the “religious leader”
Pythagoras were subjected by Walther Rathmann to so devastating
a As late as 1951 a direct attack was published: G. de Santillana and W. Pitts. “ Philolaus
εὐ τ τον; or: What Happened to the Pythagoreans?” Isis 42 (1951) 112-120. 4
Though one read in 1923 that “‘all those discoveries attributed by later writers to
| ythagoras himself were in fact achievements of certain Southern Italian mathematicians
of the time of Plato” (vi), he wrote later (Knowledge, Will and Belief: Wissen, Wollen
Glauben : Collected Essays, ed. L. Edelstein [Ziirich, 1955] p. 82) “it can hardl be doubted
that Pythagoras was the originator of this entire scientific development. He οἰ a rational
thinker rather than an inspired mystic.” Though he had in 1923 held it to be im ossibl
that Plato had taken over Pythagorean material (Philolaus) in Philebus Off if :
reads the Philebus without prejudice, one has the impression, throughout, of being in ἐς
presence of the mature result of a long philosophical career” (p. 304)— still he se
quite plainly in 1940 that Plato was dependent, in this passage, on Pythagorean ontolo ,
ess 11940] 40. Kuowledge, Will and Belief 100). On the other hand, his radical ee
cae Δ} Τα μὴν Ina review of von Fritz’s Pol,, AIP 64 (1943) 220-225.
Introduction
an analysis that scarcely anything remained, and that the keen analysis
of Harold Cherniss shook in a dangerous way the very foundation
of Zeller’s structure, that is, the reliability of Aristotle's data, it seems
that the last vestige of a possible consensus has disappeared, and it is no
wonder if resignation spreads.
The material seems to fall into the pattern each inquirer is looking
for. The historian of science rediscovers Pythagoras the scientist; the
religiously minded show us Pythagoras the mystic; he who believes in
a synthesis above rational analysis tries to show that in Pythagoras the
coincidentia oppositorum is comprehended in a Basic Idea; the anthropolo-
gist finds ‘“‘shamanism’”’; and the philological scholar may play off
against one another the contradictions of the tradition, so that critical
virtuosity may sparkle over a bog of uncertainty. Pythagoreanism is
thus reduced to an impalpable will-o-the-wisp, which existed every-
where and nowhere.
The very life of philology is the struggle between the tendencies
toward faith in the tradition and skepticism of it. In the case of Py-
thagoreanism, to be sure, the skeptical tendency has from the outset
powerful arguments on its side. The apparent primary sources, writings
of Pythagoras and his pupils, some of which are preserved while
others are known to us through secondary reference, are with very
slight exception unquestionably apocryphal. There is no longer even
any discussion of their authenticity, except for the Philolaus fragments
and some of the Archytas material.2 Of course, “forgery’’ has its
own importance in intellectual history; but in the discussion of early
Greek thought there is no place for that which—like the book of
Ocellus or of Timaeus of Locri—is obvious imitation of Platonic and
Aristotelian material.
In addition, the secondary sources, comprising the tradition about
the life and doctrines of Pythagoras and the history of his school,
contain a good many sheer impossibilities?” and, still more often,
undeniable contradictions on very important questions—whether
Pythagoras was present at the revolt in Croton, whether the numbers
6 See now Thesleff, Texts (245 pages!); as against 94 lines of text from Archytas, which
Iiels regarded as genuine, there are 46 pages of ps.-Archytas material.
27 E.g., Pythagoras was taken prisoner in Egypt by Cambyses (525 B.c.), stayed 12 years
in Babylon, and then returned to Samos in time to leave the island, in 532 B.c., because of
the tyranny of Polycrates (cf. below, ch. ΠῚ 2, α. 16), He met Phalaris about 570, and was
the teacher of Empedocles, who was born about 490. The musical experiments which are
attributed to Pythagoras are physically impossible (below, ch. V 1).
INTRODUCTION
should be thought of as corporeal or incorporeal, whether Pythago-
reans believed that the earth moves or not.28 Each side of each question
is advocated by ancient authoritics, sometimes with explicit polemic.
Even the most extreme credulity comes to a limit here; it is simply
impossible to accept “τῆς Pythagoras of tradition,” because there is no
single tradition.
Finally—less striking but scarcely less disquieting—in many cases
late tradition gives the name of Pythagoras, where older tradition,
dealing with the same topic, does not do so.?® This circumstance is
cven more suspicious when a number of apparently ancient testimonies
crumble at the touch of analysis—the name of Pythagoras is inter-
polated in Aristotle’s Metaphysics,° added by Iamblichus in a passage
from Aristotle in his Protrepticus,® supplied by Proclus, after Iamblichus,
in a fragment of Eudemus,** inserted by Porphyry in a passage stem-
ming from Theophrastus.3? When observations of this kind multiply,
Zeller’s suspicion about the “expansion of the tradition” seems
justified.
On the other hand, criticism is always subject to counter criticism.
Though many sources may be late and not very reliable, more must lie
behind them all than a simple zero. “Pythagoreanism without Pythag-
oras,” without chronological position or a place in the history of
thought, is not only unsatisfying to the scholar, but impossible in
itself. A minimalism that eliminates every aspect of tradition which
scems in any respect questionable cannot help giving a false picture.
These very difficulties in evaluating the Pythagorean tradition reveal
certain characteristic differences between Pythagoras and the other
pre-Socratics, which inevitably set the direction for a new study of the
problem. Just as a city which was continuously inhabited over a period
of time, by changing populations, presents to the archaeological
investigator far more complicated problems than a site destroyed by a
single catastrophe and then abandoned, the special difficulty in the
study of Pythagoreanism comes from the fact that it was never so dead
as, for example, the system of Anaxagoras or even that of Parmenides.
When their systems had been superseded and lost all but their philologi-
“4 Cf, below, ch. {1 2, nn. 46-47; 13, n. 52; ΠῚ τ, nn. 67ff.
Ὁ Cf below, ch. 1 2, ἢ. 76; ΠΟΙ 1V 1, nn. 19, 32, 38; ΝΠ, nn. 34-37.
CH below, ch. 1 2, n. 6.
“Of above, mot.
YOY below, ch. VE τ,
Por, Abst, 2.28, p. 158.5 14 Naucky cf J. Bernays, Uheophrastos’ Schrift tiber die
Frommigkeit (Berlin, 1866) blot.
Introduction
cal and historical interest, there still seemed to be in the spell of Pythag-
oras’ name an invitation to further adaptation, reinterpretation, and
extension. And at the source of this continuously changing stream lay
not a book, an authoritative text which might be reconstructed
and interpreted, nor authenticated acts of a historical person which
might be put down as historical facts. There is less, and there is more:
a “name, which somchow responds to the persistent human longing
for something which will serve to combine the hypnotic spell of the
religious with the certainty of exact knowledge—an ideal which
appeals, in ever changing forms, to each successive generation.
Scholarship cannot succumb to this spell. Its first task must be,
since the original phenomenon cannot be grasped directly, to interpret
interpretations, to single out and identify the different strata of the
tradition and to look for the causes that brought transformation to the
picture of Pythagoras.*4 Zeller, in his day, had seen the task and solved
it in his own way, blaming neo-Pythagoreanism for all the “‘late”’
distortions. Since the results of source analysis are hardly compatible
with this thesis, inquiry must take a new start.®> Perhaps the chances
of success are better thap before; our knowledge of the development
of fourth-century philosophy has grown, thanks to studies of Academic
and Peripatetic tradition, notably by Werner Jaeger? and Fritz
Wehrli,® and to the elucidation of the Platonic system of first principles
carried on, after Julius Stenzel, by Hans Joachim Kramer and Konrad
Gaiser. The flood of works on the pre-Socratics has provided us with
means of distinguishing fifth-century thinking, in the wake of Par-
menides, from later philosophy. And in relation to this, we should be
able to identify an even earlier stratum.
At the same time, the intention in what follows is to give its full
value to each of the various aspects of the Pythagoras tradition. One
34 The Pythagorean movement still lingers on today. The book of Jean Mallinger,
Pythagore et les mystéres (Paris—-Brussels, 1944) is dedicated to the “revered head of the
Pythagoreans of Belgium,” and his conviction is that “ Pythagoras answers, in an amazing
way, all the questions and all the needs which today’s anxiety brings forth” (p. 7). The
sane author has written a Note sur les secrets ésoteriques des Pythagoriciens (Paris, 1946).
In July and August, 1955, there was held in Brussels, Athens, and Samos a “‘ Pythagorean
World Congress”’ (cf. Schottlaender 333). Since then, Tigani, the site of ancient Samos,
has been renamed Pithagorion.
“© The task of disentangling the various branches of the tradition has been emphasized
especially by Olof Gigon; see Eutretiens sur Vantiquité classique 1 (Vandoeuvres—Geneva,
19§2) p. 141.
0. Jaeger showed how the Old Academy and Peripatos projected their own ideal, at
cach period, upon Pythagoras (Arist. φῦ, SBBIn 1928, 395f, 415ff).
7 Cf his commentary on the relevant fragments of Aristoxenus, Dicacarchus, Hera-
clides, and Clearchus.
INTRODUCTION
can take science seriously as science and at the same time comprehend
the meaning and function of that which was present before science, and
which continues to influence it. One can acknowledge the structural
unity of philosophy and still recognize that the history of thought is not
exhausted in the reconstruction of doctrinal formulae. It is inherent
in the forms of life, it cannot be separated from the unique individual,
and it cannot be repeated.
Most studies of Pythagoreanism have dealt with only one restricted
aspect; even Zeller confined himself to the development of philosophi-
cal concepts, left mathematics aside, and bracketed out religious and
ethical questions; and later works have been even more specialized,
whether in the philosophical area, in that of mathematical, astronomical,
and musical problems, or that of religion. The very thing that might
seem rash and hasty, in view of the fundamental differences of inter-
pretation, is what the nature of the situation demands: as many-sided
a treatment of the problem as is possible. For many of the contradictory
conclusions have come from investigating and tracing the course of
single paths of development, with no thought of the way in which
these may converge with other, equally important lines. Any attempt
to date Pythagorean philosophy and science back as far as possible,
even to Pythagoras himself, must take account of its connection with
the religious-cultic and primitive sides attested for the movement; and
any investigation of Pythagoras the ““shaman’’ must endeavor to make
sense of the later development of Pythagorean science. The division
of labor which seems so reasonable brings along with it the danger of
a vicious circle. It can happen that the historian of science builds his
reconstruction on a philologically inadequate foundation; the philolo-
gist takes over the seemingly exact result of the historian of science;
the philosopher, on the basis of this criterion, rejects contradictory
evidence—and so on. The many-sided treatment which can prevent
misunderstandings of this kind must be the objective of the classical
scholar; collection, interpretation, and critical analysis of the ancient
evidence is the necessary common foundation. Mathematical and
philosophical thought, as well as parallels from comparative religion,
can never yield more than possibilities; as to the historical facts, the
sources are decisive.
At the beginning of the following studies stands, not the analysis
of the oldest evidence, but discussion of the tradition of Pythagorean
philosophy ; the task turns out to be not to separate older and more
Introduction
reliable from later and more dubious material, but to judge between
traditions which are contradictory but of equal antiquity. In this matter
a remarkable stroke of luck has provided us with an important new
piece of source material, a fragment of Speusippus on Pythagorean
philosophy first published in 1953. This makes certain, what a careful
analysis of the sources would in any case make likely, that a Platonizing
interpretation of Pythagorcanism, which had a decisive influence on the
later tradition, gocs back to Plato’s immediate disciples and differs
sharply from the reports of Aristotle. The latter’s evidence thus be-
comes more important than ever; for he alone warns us to separate
Pythagorean and pre-Platonic from Platonic material. Only from the
point of view gained by the revelation of this contrast is it possible to
evaluate Plato’s own testimony, and in particular the allusions in the
Philebus. In this way we find incontrovertible evidence—as against
Frank and Howald—of a pre-Platonic and pre-Socratic philosophy of
the Pythagoreans.
Study of the oldest, pre-Platonic tradition can thus be supplemented
by those pieces of evidence which stand outside the Platonic influence,
and were not affected by the reinterpretations mentioned above. Once
more the reports of Aristotle become especially important, the frag-
ments of his lost monograph on the Pythagoreans. The Pythagoras
story, which used to be, for the most part, written off as the unfortu-
nate product of the obfuscation of historical facts, may be understood as
the expression, precisely, of a definite historical reality. Pioneers of this
line of interpretation were Karl Meuli and E.R. Dodds. To the legend
belong the acusmata, in their essence doubtless extremely old. The
result that emerges is a rather distinctive picture of a shamanistic
“ Wise Man”’ and a Life, or Way of Life (bios), dominated by ritual—a
Pythagoreanism foreign to all exact science.
The study of the Philolaus fragments harks back to the results of
the first chapter. A point of view is achieved from the distinction of
Platonic and Aristotelian traditions about Pythagoras, from which one
can sec clearly the genuineness of at least part of these fragments. The
attempt is made, then, to make these understandable in the framework
of fifth-century thought, as the attempt of a Pythagorean to come to
terms with the natural science (physiologia) of the Ionians and the
Eleatics.
The history of the natural sciences can confirm this result, for on
one hand the Philolaus testimonia take their position in the history of
fifth-century thought, while on the other the origin of the exact
Ἂ
INTRODUCTION
sciences is seen to lic outside the Pythagorean realm. The renown of
Pythagoras as the inventor of mathematics and mathematical natural
science is explicable as a distortion of perspective; a pre-rational
interpretation of the cosmos, along with arithmological speculation,
is seen in hindsight as rational science. | ᾿
My aim has been to take full account of the ancient evidence ;**
and the modern literature has been consulted as fully as was practicable.
As far as possible the originator of cach particular thesis or argument
has been named, but it was out of the question to try to provide an
encyclopedic doxography of all pertinent views on each problem. I
hope nothing decisive has been overlooked.
Mt is often not necessary to cite all parallel passages. Citation of a passage im peas
οὐ in Diogenes Lacitius is always to be taken as also ἃ reference to the collection of parallels
in the editions of Deubner and Delatte, respectively.
I. Platonic and Pythagorean Number Theory
I. THE PLATONIC THEORY OF IDEAL NUMBERS
Πλάτων mvbayopile.—from the time of Aristotle, this finding has
often been repeated, but there is little clarity as to the extent and The
manner in which Plato borrowed Pythagorean doctrine, or as to what
Pythagoreanism was like before Plato. This applies especially to those
attempts to derive the Ideas from numbers, to equate them with
numbers, or even to replace Ideas with numbers, which Aristotle and
others attribute to Plato and his pupils, in particular to Speusippus and
Xenocrates. These theories looked somewhat like a capitulation to
Pythagoreanism on the part of the aging Plato, until Julius Stenzel set
the task “to understand the concepts of Plato’s late philosophy in their
own context and to incorporate into Plato’s philosophical develop-
ment, inseparably, those ‘Pythagorean mystical’ elements that have
up to now been traced back to external, foreign influence.””? Since
then, much scholarly work has been devoted to the Platonic “doctrine
of principles,”? and to its interpretation in Platonic terms, starting
from Platonic presuppositions. Still the question of pre-Platonic
sources, of pre-Platonic Pythagorean doctrine, persists. The crucial
point is to grasp firmly what the difference is between this and the
later, Platonic philosophy. An investigation must start from a sketch
of the Platonic theory of principles, since we know more about its
context.
The foundation for the study is the evidence of Aristotle; he alone
scts up Platonic and Pythagorean doctrines side by side, specifying
‘Esp. Met. 987429 ff.; cf. e.g. Aét. 2.6.6; Cic. Rep. 1.16, Fin. 5.87, Apul. Flor. 15,
p. 60, Apul. Plat. 1.3, ch. I 4, ἢ. 47.
2 ZG 108.
* Handy bibliography in the German reprint of Stenzel ZG and in Gaiser, who also
vives a convenient collection of the sources. The most comprehensive work remains
ἱ,. Robin, La théorie platonicienne des idées et des nombres Δ᾽ ἀργὸς Aristote (Paris 1908, repr.
14963). New evidence was adduced by Merlan (1934) and Wilpert (1940, 1941). The paper
of H. Gomperz (1930) deserves special mention, because it has been generally overlooked.
A new, energetic attempt at reconstruction and interpretation has been made in the books
of Kramer (19$9) and Gaiser (1963); among more recent contributions, see Theiler
fsonomia; Done; KH Mting, Phronesis 14 (1g68) 1-31.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
their points of agreement and—what is almost more important—their
differences.*
This might appear an extremely unfavorable point of departure.
Precisely as the result of the amount of recent research in this field, it
remains one of the most controversial in Platonic studies, without any
prospect of early agreement. There is controversy as to how the con-
nection of ideas and numbers is to be understood, how the whole
matter is related to the well-known philosophical theories expressed in
Plato’s dialogues, and what is to be attributed to Plato himself and what
to his various pupils. And the most learned studies in the field, the
books of Harold Cherniss,® have shaken the very foundation of the dis-
cussion with the thesis that most of the supposed evidence is wrong
inference, or even falsification, by Aristotle, and that the true Plato is
to be found only in the dialogues and has nothing to do with these
speculations. At the other extreme, Kramer and Gaiser hold that the
backbone of Plato’s philosophy is a “doctrine of principles” (Prinzi-
pienlehre) which can be found even in allusions in the dialogues. The
lively, sometimes acid, discussion which has been going on for years
will not easily be brought to a solution.®
Fortunately for the reconstruction of Pythagorean doctrine, not
much depends on the solution of these problems. What the relation is.
between Plato’s dialogues and the “doctrine of principles,” whether it
represents an early or only a late stage in Plato’s philosophy, whether
a particular doctrine is to be traced to the historical Plato, to Speusippus,
Xenocrates, or even to Aristotle’s interpretation, becomes irrelevant
when the issue is to establish the difference between Platonism in
general and earlier Pythagoreanism. In what follows, the adjective
“Platonic” may be understood as applying to “Platonists’” and
“Platonism,” regardless of the question of their relation to the philo-
sophy of Plato himself.
For, though Cherniss has rightly emphasized the independence of
4 Astonishingly little attention has been paid to Aristotle’s distinction of Platonism from
Pythagoreanism, though Zeller (I 4658) collected the evidence. Thus down to Raven
(KR nos. 320, 405, 406) and Guthrie (I 256-262; cf. below, ch. I 2-3). Platonic doctrines
are included among Pythagorean documents. Frank denied himself a “more exact
source-analysis of the Aristotelian and post-Aristotclian reports,” because of the inadequacy
of “the space available” (n. 388); thus he missed what would have refuted his thesis
(cf. below, ch. I 2) that all “Pythagorean” philosophy is post-Platonic.
®Cherniss, Plato, vol. I—the second volume is not expected to appear—and Riddle,
esp. pp. 2off. |
4 Fora polemical survey, cf. Krimer, MH 21 (1964) 137-167; Philologus 110, (1966)
48-70; earlier criticism of Cherniss’ theses: De Vogel, Mremosyne 1949; Ross, PTT 142ff.
1. The Platonic Theory of Ideal Numbers
Plato’s pupils and the differences between their doctrines,’ still it is Hot
to be denied that there is a common tendency in the area of “first
philosophy’’—to use Aristotle’s term—in their essays at basic onto-
logical doctrine. The aim is to trace ideal Being to its basic principles
(ἀρχαί) and to understand this Being in terms of these principles, just
as the Platonist learned to understand the world in which we live from
the point of view of the ideas. Heinrich Gomperz proposed the con-
venient term “system of derivation,” though this characterizes the
movement of thought in a one-sided manner. It is rather the ascent
from the empirical to the Higher, the vision of the Idea through the
world of experience that surrounds us, and the realization of the agathon,
the One, in the complexity of the ideas that Platonists strive for, than
conceptual “deduction” or “derivation” (Arist. EN 1095a32).
The problem of unity and multiplicity of ideas, of their interpene-
tration and their distinction by the method of diaeresis was, as the
dialogues show, of increasing concern to Plato. The borderline
between dialectical exercise and metaphysical seriousness seems to be
deliberately blurred. There occurs a curious reticence when the con-
versation touches upon the most essential questions, even in early dia-
logues.® In the Timaeus, Plato gives more definite indication that it is
the question of the πάντων ἀρχή (the first principle of all things) that
is being bracketed out (48c): Plato has reduced the multiplicity of the
world to the four elements, the elements to regular solids, and these to
triangular surfaces; “‘but the principles that are still prior to these god
knows, and he among men who is dear to him” (53d). The Republic
introduced the Good as the Highest, the Sun in the realm of ideas,
“beyond being” (so9b). The opposite to the good, in all the later dia-
logues, is described as an indefinite oscillation in two directions,
toward “great and small,” “more and less,” “‘stronger and weaker,”
and it is the Good that constitutes measure and definiteness in this
continuum.??
It is from this point of view that we can approach the reports about
Plato’s oral teachings," in particular the lecture On the Good. Our
7 Riddle 60ff.
* Parm. 129aff (and passim), Phib. 15a, Soph. 251aff. Cf. Kramer 420ff.
® Prot. 357b. Cf. Kramer 1806
This has been worked out in a convincing way by Krimer.
''In one passage Aristotle cites explicitly the “unwritten doctrines” of Plato (Phys.
2oybis; cf. Cherniss, Plato 113ff, 166f, Kramer 416ff), and he clearly refers to them in
other passages (Ross, PTT 143%); once (gg2a22) he speaks of Plato's “repeated” pro-
houncenients,
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
principal source for these is the comments of Aristotle.!? Besides,
written records of this lecture by Spcusippus, Heraclides, Xenocrates,
and Hestiaeus are attested,!3 and it is possible that a detailed report in
Sextus Empiricus was derived, by a roundabout route, from one of
these.!4 Simplicius has preserved an important fragment from a report
of Plato’s pupil Hermodorus;! besides, Aristoxenus reports, on the
basis of an oral account by Aristotle, on the external circumstances and
the content of the lecture, and there are some relevant statements in
Theophrastus’ Metaphysics. Reconstruction may proceed here on a
comparatively broad basis. It is hard to get rid of the tradition which
traces the theory of ideal numbers to this lecture;!* proceeding from it,
Plato’s students developed the theory in their own way. In reporting
on it, Aristotle’s attitude varies according to the subject he is dealing
with. The ideal numbers are not treated in Περὶ i8edv,!® whereas in
12 Fry, 27-31, on which see esp. Wilpert, Hermes 1940. Even Cherniss concedes that
Alexander of Aphrodisias used the book directly, though obviously no one did so after
him (Plato 119n. 77; Riddle 2786).
13 Simpl. Phys. 151.6ff, 453-25ff; for Heraclides, D.L. 5.87; for Xenocrates, D.L. 4.13.
14 Sext. Emp. Math. 10.248-284. Cf. ch. I 3.
15 Phys. 247.30ff; see esp. Wilpert, Hermes 1941, 227ff, and Zw Fr 183ff; De Vogel,
Mnemosyne 1949, 205ff; Cherniss, Plato 285ff; Kramer 2828.
16 Harm. 2, p. 30 M.: the logoi had to do ‘“‘with mathematics, numbers, geometry,
astronomy, and, finally, that the good is single” (translating Macran’s conjecture
<1 >dyaGev and taking τὸ πέρας as adverbial. Cf. Cherniss, Riddle 87 n. 2, Ross, PTI 148
n. 1, 244, Kramer 423; differently interpreted in De Vogel, GP I 274 n. 1).
17 Cherniss tries to undermine this foundation with the observation that in the reports
about the lecture On the Good the point corresponds to the One (Alex. Met. §5.20ff;
Simpl. Phys. 454.19ff; also Sext. Emp. Math. 10.259ff), while according to Aristotle
Plato eliminated the concept of the point (992a20ff; Cherniss, Plato 167ff, Riddle 28f).
De Vogel’s attempted compromise, Mnemosyne 1949, 306ff, is not completely convincing.
In the passages of Aristotle, however, which refer to Plato, there is no mention of the
point as a “‘monad having position”; the first number is 2 (cf. 1081a21ff), and the first
geometrical entity is the line, whereas Speusippus adopts the point corresponding to One.
Thus there is some inexactitude in the later reports (cf. Kramer 418 n. 76: the commenta-
tors are using mathematical terminology). This tends to confirm the reliability of Aristotle.
18 While Cherniss seeks to isolate the lecture On the Good as much as possible (Riddle
12), Ross speaks of a ‘course of lectures” (PTI 148), and Kramer states: “The discourses.
on the Good provide ...the standard expression for the teaching activity of Plato in
general” (409); he could refer to πολλάκις Pl. Ep. 7.3424 and Arist. Met. 992422, but his
argument from the “imperfect of repetition” (407) in the passage from Aristoxenus
(Harm. 2 p. 30 Μὰ.) is not conclusive; he would understand “every time when the lecture
turned out to be on mathematics... this would seem, I think, paradoxical to the audience.”
But ὅτε δὲ φανείησαν may as well be optative in indirect discourse (W. Theiler, AGP $0
[1968] 29 n. 1). Aristotle used to tell the anecdote as a warning how not to proceed in
lecturing, how to avoid disappointment by correct advertising. Evidently not the audience
alone had been disappointed, Thus this experience has—against Kriimer—nothing to do
with the ‘test’? described at Ep. 7.340b, the deliberate discouragement of merely curious
auditors by demonstrating the difficulty of philosophy,
Theiler, sonomia g2, draws chronological conclusions from this. That the theory
of ideas was first set up without recourse to ideal numbers is stated by Aristotle at Met.
1Ο͵ΆΪ 1,
1. The Platonic Theory of Ideal Numbers
his discussions of “first philosophy,” he has a tendency to put a dis-
proportionate cmphasis on number theory: “If the Ideas are not
numbers, it is impossible for them to exist at all; for from what kind
of principles will the Ideas come?” (1081a126).
The most important hints at these unwritten doctrines in the corpus
Platonicum are to be found in famous passages of the seventh Letter;
no wonder therefore that the discussion about the “esoteric”’ Plato is
intimately bound up with the question of the genuineness of that
letter.2° Again, however, the answer is of limited importance for the
investigation undertaken here: it is generally agreed that the letter is
either by Plato, or by one of his immediate pupils, and, even in the
latter case, the doctrines contained, especially the “ philosophic digres-
sion’ have about the same degree of authenticity as the reports on the
lecture On the Good.#* Dionysius, the letter says, has learned from
Archytas (338c) and Dion something of the doctrines which Plato him-
self had not yet wanted to communicate to him, parakousmata (338d),
chance bits of information about “the greatest matters” (341b),
“something of that which is highest and first in the philosophy of
nature’ (344d; cf. 341d). It concerns “truth about virtue and vice”
(349a-b), and at the same time “truth and falsity in the whole of being”
(344b), one singular object of learning (mathema), which is not easy
to communicate (3410). The “greatest teaching” is the Good (Rep.
5058), thus the close relation between the comments in the seventh
Letter and the later memoranda of his pupils On the Good is obvious.
As the audience was startled that “the Good” should concern mathe-
matics and astronomy, the letter binds up “philosophy of nature”
with “ virtue and vice.” Plato states however, “There is not any writing
of mine on these matters, nor will there ever be, for this is a thing
which cannot be put into words like other doctrines” (341c)—a
sentence as famous as it is controversial in its interpretation. While
Cherniss believes that anyone who takes these words as genuine and
Ὁ The authenticity of the seventh Letter had been a cornerstone of Platonic scholarship,
especially in Germany, since Wilamowitz’ Platon (1918), but Cherniss pronounced a
negative verdict (Riddle 13), and proofs of inauthenticity were undertaken by G. Miiller
Archiv Philos. 3 (1949) 251-276 (contra, H. Patzer, ibid. 5 [1954] 19-36, and B. Stenzel.
AJP 74 [1953] 383-397)—and above all by L. Edelstein, Plato’s Seventh Letter (Leiden
1966) (contra, K. von Fritz, Platon in Sizilien, [Berlin, 1968]}). The aid of the computer has
been invoked: M. Levison, A. Q. Morton, A. D. Winspear, Mind 77 (1968) 309-325;
but to attribute not only the seventh Letter but the introduction to the Timaeus and the
Critias to Speusippus can hardly be the final solution.
*!'The philosophical digression is introduced as a λόγος, πολλάκις ... ῥηθείς
(3424). For interpretation, cf. HG, Gadamer, “Dialektik und Sophistik im siebenten
platonischen το SBHeid 1964, 2; K. von Fritz, Phronesis 11 (1966) 117-153. On
ὄνομα, λόγος, εἴδωλον, Pl. Pol, a85c-286a; οὐσία, λόγον, ὄνομα, Leg. 8ygd-e.
΄
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
relates them to the On the Good must in consistency give up trying ever
to understand Plato, Krarmer emphasizes that these “‘ ultimate matters”
are not “inexpressible” in an absolute sense, but only for the great
mass of mankind.22 Yet Plato himself shows how serious he is about
this “‘inexpressibility” by tracing its cause, in an excursus, to the rela-
tion between Being and the means of knowing. There are four means
or steps of “knowing” an object: by name, by definition, by image,
or by knowledge to which mind and right opinion are added (ἐπιστήμη
καὶ νοῦς ἀληθής τε δόξα, 342c). But the fifth, Being itself, stands apart as
“that which is the object of knowledge and truly exists” (342b). Mind
(nous) comes closest to this (342d) but even mind does not grasp it
completely and unambiguously. Each of the four kinds of knowing
comprehends a qualitative aspect (poion ti) as much as the Being which
the soul is seeking (342e) :38 “‘ There are two things, the Being and the
qualitative aspect, and it is not the “what kind’ but the ‘what’ that the
soul seeks to know. But each of these four proffers to the soul the thing
that is not being sought, and thus fills every man’s mind with puzzle-
ment and unclarity’’ (343b-c). Because of this inadequacy—even on
the part of mind and knowledge—it is easy to contradict and refute.
where the “fifth” is concerned: ‘‘ When we are under the necessity of
separating out and revealing the fifth element, anyone who likes to do
so has the means of confuting us” (343d). The ring of disappointment
at the “contempt” (341e, 3450) to which the most sublime is exposed
may echo the contempt (ὑποκατεφρόνουν Aristox.; above, n. 16) shown
by hearers of the lecture On the Good. Therefore the only course
possible, the letter says, is to take those who are already on the right
path and are “related to” truth (344a), and lead them on, in patient
practice, to the point where Being reveals itself to them in a sudden
illumination: “Knowledge of each thing, and Mind, blaze forth in
his view as he strains to the limit of human capacity” (344c).4
If this is taken seriously as Plato’s view, the dialogues are devaluated
as lacking seriousness, being only preliminary guideposts to what ought
22 (Cherniss, Riddle 13; Kramer 25ff, 401, 457ff; Gaiser 4f. According to Pl. Leg. 968e,
the most important mathemata are not secret, (ἀπόρρητα), but cannot be told in advance
ἀπρόρρητα).
Ὁ The ἡ ΠΕΣ τί- ποῖον appears as carly as Gorg. 4485, Meno 71b. To modern logic,
the τί question scems to be almost void of sense (I. Robinson, Plato's Earlier Dialectic
[Oxford, 1953] 49-53) which shows that it is not modern logic for which “the soul is
longing.” According to ‘Tim. 4gd (cf. Arist. Met. 1033b21), the object of the senses, as
contrasted to the idea, is only a ποῖόν τι, never a τί, but this statement is not identical
(against Wilamowitz, Platon I 294; Kramer 304, 459 0. 15 5) with the doctrine of the
seventh Letter, where even ἐπιστήμη cannot grasp the τί,
"ὁ συντείνων MSS, em. E. Sachs (Wilamowitz, Platon Il agg a. 2): συνταίνοντι.
1. The Platonic Theory of Ideal Numbers
to happen in a philosophical life. But also, the oral teaching can never
be a final “system,” made up out of non-contradictory rules and
terms. For not even knowledge firmly grasps its object. Plato is not
satisfied with elegant “principia mathematica,” he asks for the Good—
not in the sense of an ‘“‘ought”’ imposed by some authority, but as the
goal of all striving, the prop and the meaning of existence. This can
only be perceived in an individual experience similar to religious reve-
lation. But the only way to it is the common quest of indefatigable
dialogue, which necessarily leads to the most stringent rules of dialectic,
to logic and mathematics. There will be no final result, which can be
taken down on paper or papyrus, so that any written account must be
misleading. When Plato’s students wrote about “the Good,” they
deviated from this attitude ;?> but it was doubtless a necessary task, once
the discussion had grown into a variety of conflicting interpretations,
to put down what Plato himself had taught. Independent philosophiz-
ing passed over very quickly, in the Academy, into history of philosophy
—interpretation of the authoritative thought of the dead Master.?6
For the historian, it is exact doxography that matters; so what follows
is an outline of doxai on principles attributed to Plato. The highest
principle of Platonic ontology is the One; alongside 157 stands the
Indefinite Dyad, a principle that is also described as great-and-small,
many-and-few, exceeding-and-exceeded, and unequal.?8 It is respon-
sible for every kind of multiplicity, contrast and change in the realm
*5 Kramer (412) thinks that the seventh Letter would allow ὑπομνήματα in the sense of
the Phaedrus (276d; 2784), but 344d expressly rejects ὑπομνήματα. Ross, PTI 158, on the
other hand, thinks that Plato forbids prose writings, and that the semipoetic dialogues
are left untouched; but 344c5 seems to allude to the main works of Plato.
#6 Aristotle, of course, is the exception; he combined criticism with his notes. Fr. 27:
‘‘one ought to remember that we are but men [i.e. subject to failure] not only in the
pursuit of happiness, but also in carrying out a demonstration.”
*? There is no word about any derivation of the “Indefinite Dyad”’ from the One; the
later, so-called Pythagorean tradition presents monistic as well as decidedly dualistic
interpretations, cf. below, ch. I 3.
“8 Arist. fr. 28 = Alex. Met. 56.8ff, Simpl. Phys. 453.33ff; ἡ τοῦ dvicov Suds, Met.
1087b7; μὴ ov, Phys. 19247. There was some doubt whether the term ἀόριστος δυάς could
be traced to Plato himself (Ross, PTI 184; Met. 1169), until the new fragment of Speu-
sippus (Plato Latinus ΠῚ 38; below, ch. I 3) was published. Since Speusippus himself called
the second principle πλῆθος, it is from older tradition that he took the term “‘inter-
tninabilem dualitatem,” i.e. from Plato. Aristotle expressly refers to the ‘“‘unwritten doc-
trines’’ for the concept of μεταληπτικόν (μέγα καὶ μικρόν), Phys. 209b16. Epin. g90d—991b,
4 passage intentionally obscure, alludes to the role of the Dyad (on this passage, A. R.
lacey, Phronesis 1 [1955-1956] 81-104). Ross (PTT) tries to render the concept by “‘bare
plurality” (which, strictly speaking, is rather Speusippus’ πλῆθος); Becker (ZwU 18) by
“logical extension,” though Plato wants us to think of this “extension” as a duality, a
deviation in two directions fron: the center, the measure, the One. Krimer has shown
that this pattern of thoughe is discernible in ethical discussions of the late dialogues
(140lT, 244ff).
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
of Being, as against the unity, identity, constancy brought about by
the One. The One is identical with the Good ;29 the Indefinite Dyad is
the ground of all evil. It is also called Not Being (Arist. Phys. 19247).
In Aristotle’s terminology, the two principles are also related as form
and matter.
Both principles “beget’’®° numbers from themselves. First 2 comes
to be, as the “indefinite dyad”’ is limited by the One and transformed
to the definite number 2 which consists of two equal units.*! According
to Aristotle, the other numbers arise in the natural succession 3, 4, 5,
- etc.;32 but Aristotle emphasizes the difficulty—nay, impossibility—of
deriving in this way other numbers than those of the type 2".38 The
numbers that arise in this way are independent entities, which cannot
be combined in arithmetical calculations; their units are οὐ συμβλητοί,
as Aristotle puts it.34
The relation of these numbers to the mathematical numbers which
are used in calculation is hard to establish. Speusippus and Xenocrates
put forth different solutions for this question: the former eliminated
ideas and only recognized mathematical numbers as the ground of
reality; Xenocrates equated ideal and mathematical numbers.** The
main concern of these philosophers, however, is not to lay the founda-
tions of mathematics, but to explain the world by means of its princi-
ples. The ideal numbers are not only the ideas of particular numbers—
“twoness,” “threeness,” etc., but somehow govern the structure of
reality: they are ideas themselves. It is not clear how this connection
29 This is why the lecture was entitled On the Good; cf. Aristoxenus (above, n. 16),
Arist. Met. 988a14, who, according to his own philosophy, says Plato “assigned _the
Good to the One, as if this were a second step, whereas for Plato there was basic identity.
30 γεννᾶν is used as a technical term, notwithstanding its mythological connotations.
The ontological priority is established according to the principle ἀναιρουμένου τινὸς μὴ
συναναιρεῖσθαι: of two concepts, the one which can be thought without presupposing the
other is “prior”; thus number antecedes geometrical magnitudes, the line antecedes the
plane, etc. (101ga1; Alex. Met. $5.22).
81 1981a23, 1083b23, 1091424, fr. 28 = Alex. Met. 56.8-35.
32 1980024, 1081021, 1081b30, 1080a33, Ross, PTI 191. |
33 :oy1alo. The Indefinite Dyad “produces duality” 1082a15; on the generation of the
number Four, 1o8tb2t, 1082413. At 987b33 Aristotle says numbers can be “easily”
generated out of Indefinite Dyad, ἔξω τῶν πρώτων, a much debated expression (Wilpert,
Zw 207; Ross, PTT 188; Becker, Zw 8), since the meaning “ prime number does
not fit 10gtaro. “The One is said to be the cause of the Odd (1084436), falling into’
the even number (1084a4), constituting the middle of even numbers (1083b28). Thus it
would seem that the series of integers is produced by doubling and by adding one (cf.
Alex. Met. $7.24), and still Aristotle states there canbe no addition in the sphere of ideal
numbers (πόδα 2, 35, 1o80Oa23; Ross, PE τ cf Wilpert, Ζιν ιν 21 4tF).
M8 OROAIT: τ08 1}01ς,
85 'The evidence is collected by Ross, PVE ςας
1. The Platonic Theory of Ideal Numbers
of ideas and numbers is to be understood, in detail. While Aristotle
says simply that the ideas are numbers, Theophrastus speaks of an
“attaching” (anaptein) of the ideas to certain numbers, and thus allows
us to imagine a looser relationship.** The attempt of Stenzel and
Becker to understand the ideas as “numbers” on the basis of the
method of division (diaeresis) prominent in Plato’s late dialogues is not
convincing.*’ .
The most familiar example, the first step in the derivation of the
world from numbers, often cited by Aristotle, is the sequence of
geometrical dimensions: the line corresponds to the number 2, the
plane surface to 3 (the triangle being the simplest conceivable plane),
the solid (tetrahedron) to 4. It is recognized that Speusippus®* and
Xenocrates*® worked out this derivation. Speusippus has the point at
the beginning as corresponding to the One.‘ Aristotle also quite
clearly attributes to Plato the derivation of line, plane, and solid “after
the numbers”’;*! here the “‘unlimited dyad,” the “great-and-small,”
appears as “short and long,” “broad and narrow,” “deep and
shallow.’’42 One text derived from the lecture On the Good*® tells us
that the numbers 2, 3, 4 correspond to this derivation, but the testi-
mony is unreliable insofar as, like Speusippus, it has the point corre-
sponding to the One.“ That Aristotle attributed to Plato himself the
38 Ross, PTI 216 ἢ. 1, has collected the passages of Aristotle; Theophr. Met. 6b11, who
is followed by Ross 218: Plato “assigned numbers to ideas,” without identification; other
attempts at interpretation: De Vogel, Mnemosyne 1949, 311; Wilpert, ZwFr 170.
? After Stenzel’s vague suggestions, Becker tried to give a precise solution in QSt
1 (1931) 464-501, defended ZwU; criticism in Cherniss, Riddle 54ff; Ross PTI τος.
The main argument against Becker is that the “ideal number” would change with
cvery new definition of an idea, and the same number would be attributed to quite
different ideas. It is true that Alex. Met. $7.6 says that διαίρεσις is γένεσις ἀριθμῶν; but the
number 2 is not generated by diaeresis (above, ἢ. 33), nor is the sequence line~plane-solid
a diaeresis.
“Fr. 4 = Th. ar. 84.10 πρώτη μὲν yap ἀρχὴ εἰς μέγεθος στιγμή, δευτέρα γραμμή, τρίτη
ἐπιφάνεια, τέταρτον στερεόν.
" Pr. 39; cf. fr. 34 = 1028b24.
Δ 1085232: ἕτεροι δὲ ἐκ τῆς στιγμῆς (γεννῶσιν τὰ μεγέθη)" ἡ δὲ στιγμὴ αὐτοῖς δοκεῖ
εἶναι οὐχ ἕν ἀλλ᾽ οἷον τὸ ἕν, cf. Top. 108b7, b26; Cherniss, Plato 131 ἢ. 82.
1 This is clear from the passages where questions are raised about the relation of the
ideal magnitudes so produced to the “intermediate” realm of mathematical magnitudes
(Qu2bi3ff, 1o80b23ff, 1085a7ff). Since neither Speusippus nor Xenocrates accepted this
intermediate realm, the reference must be to Plato (Ross, PTI 206ff).
Δ oo2aloff, 1085a7ff, 1088b4ff, to8ybriff, Περὶ dir. p. 78.20ff Ross.
4 Sext. Emp. Math. 10.278ff; Alex. Met. §5.18ffand Simpl. Phys. 454.19ffdo not mention
this relation to the numbers.
"See above, ἢν 17. At 108$a7ff, Aristotle distinguishes between those who begin
with the “great-and-sinall” and “others” (Speusippus: above, n. 38) who begin with the
prorat.
I, PLATONIC AND PYTHAGOREAN NUMBER THEORY
derivation of the line from 2, the plane from 3, and the solid from 4,
seems probable on the basis of two passages, though both are very
controversial.4® The well-known reduction of the physical world in
the Timacus, adequately evaluated by modern natural scientists,
seems to find here its natural sequel. If the elements are traced back to
polyhedra, and polyhedra, the simplest of which is the tetrahedron, to
triangular planes, and the question of their “origins,” the “more
ultimate principles” (ἀρχαὶ ἄνωθεν) is explicitly excluded (53d), the
reduction in the present passage leads further, to the line and finally
to the ultimate principles, the One and the Indefinite Dyad.
Aristotle further explicates the doctrine of ideal numbers in connec-
tion with psychological theory: Plato is said to have formed the soul
out of “elements,” in the Timaeus, in pursuance of the thought that
like is known by like;*
45 (a) At 10gob20ff, after his criticism of Speusippus (b14ff), Aristotle turns his attention
to those “‘who posit ideas” (of ras ἰδέας τιθέμενοι): ποιοῦσι yap τὰ μεγέθη ἐκ τῆς ὕλης
καὶ ἀριθμοῦ, ἐκ μὲν τῆς δυάδος τὰ μήκη, ἐκ τριάδος δ᾽ ἴσως τὰ ἐπίπεδα, ἐκ δὲ τῆς τετράδος
τὰ στερεὰ... ἀλλὰ ταῦτά γε πότερον ἰδέαι ἔσονται... καὶ τί συμβάλλονται τοῖς οὖσιν;
οὐθὲν γάρ, ὥσπερ οὐδὲ τὰ μαθηματικά (cf. b13) οὐδὲ ταῦτα συμβάλλεται. ἀλλὰ μὴν οὐδ᾽
ὑπάρχει Kar’ αὐτῶν οὐδὲν θεώρημα, ἐὰν μή τις βούληται κινεῖν τὰ μαθηματικὰ καὶ ποιεῖν
ἰδίας τινὰς δόξας... οὗτοι μὲν οὖν ταύτῃ προογλιχόμενοι ταῖς ἰδέαις τὰ μαθηματικὰ
διαμαρτάνουσιν" of δὲ πρῶτοι... (and here follows criticism of the connection of ideal
number and mathematical number in Plato). This passage has most often been interpreted
as applying to Xenocrates (fr. 38 Heinze; Ross, Met. 2.481; Cherniss, Plato 568, Gnomon
1959, 45ff; De Vogel, Mnemosyne 1949, 303). On the other hand, Ross (PTI 208f) and
Saffrey (25ff) hold that Xenocrates is referred to individually only from ἐὰν μή Tes On,
and that in the earlier part both Xenocrates and Plato were meant. Ross’s argument that
in b26 the “mathematicals” are the “intermediate realm” of Plato and not of Xenocrates,
will not hold up against Cherniss (Gnomon 1959, 47), who has recognized here an allusion
to the previously discussed theory of Speusippus. Saffrey argues that the question whether
line, surface, and solid are ideas or a separate class must be asked with reference to Plato,
because Xenocrates equated ideas and mathematicals. Aristotle poses the same question
at 992b13ff (cf. above, n. 41), with an unambiguous reference to Plato. The clause οὐδ᾽
ὑπάρχει ye κατ᾽ αὐτῶν οὐθὲν θεώρημα cannot be directed against Xeriocrates, who was
precisely the one who wanted to force ideas and mathematicals together (προογλίχεσθαι).
Thus after all the interpretation of Ross and Saffrey is preferable.
(b) In 1036bi12ff, the question is discussed: What belongs to the Form (eldos)? Do
flesh and bones belong to the Form of Man? Does the line, or the continuous, belong to
the Form of triangle or circle? Some deny this, καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, Kat
γραμμῆς τὸν λόγον τὸν τῶν δύο εἶναί daow" Kai τῶν τὰς ἰδέας λεγόντων οἱ μὲν αὐτογράμμην
τὴν δυάδα, οἱ δὲ τὸ εἶδος τῆς γραμμῆς, ἔνια μὲν γὰρ εἶναι ταὐτὰ τὸ εἶδος καὶ οὗ τὸ εἶδος
(οἷον δυάδα καὶ εἶδος δυάδος), ἐπὶ γραμμῆς δὲ οὐκέτι... . Once more he deals first with Speu-
sippus, who did not accept any Ideas, and then distinguishes two directions in the theory
of Ideas, and the probability is that they are those of Plato and Xenocrates. Cherniss
(Plato $67, Gnomon 1959, 44) takes αὐτογραμμήν, because it lacks an article, as predicate
noun with both δυάδα and τὸ εἶδος τῆς γραμμῆς, appealing to “the elementary rules of
Greek grammar.” These do not apply, however, in technical philosophical language
(cf., just previously, b13: γραμμῆς τὸν λόγον τὸν τῶν δύο εἶναι). The one named first, he
thinks, Xenocrates designated the dyad, while Plato designated the “idea of line” as
“Line in itself”; thus the doctrine of ideal numbers and the ordinary theory of ideas stand
1. The Platonic Theory of Ideal Numbers
and in the same way it was laid down in the work entitled On
Philosophy that the Animal-itself is composed of the Idea itself of the
One, along with the primary length and breadth and thickness, and
the rest in a similar manner. Again, putting it differently, mind was
said to be the One, knowledge two (because it goes in a straight line
to the One), whereas the number of the plane figure is opinion and
that of the solid, sensation ... And, since the soul seemed to be
productive of both motion and knowledge, some have compounded
it of both . . . (De anima 404b18ff).
Is it a doctrine of Plato*® or of Xenocrates*® that is reported here?
In the expression “in the work entitled On Philosophy” (ἐν rots Περὶ
φιλοσοφίας λεγομένοις) Aristotle is referring to his own dialogue,>°
which therefore developed further the hints of the Timaeus (53d) about
the “‘first principles of things”’ (404b17), and included mention of the
series of dimensions.*! But the expression “again, putting it differently ”’
side by side. In this case, however, the explanation beginning μὲν yap must be understood
as applying to the first instead of the second solution (Gnomon 1959, 44 n. 3), and the
phrase beginning of δέ is not only a tautology but destroys the continuity. In 1043233ff,
where the same problem is treated, there is a consistent connection of dyad and line.
Therefore the translation must be, “Of those who accept the ideas, some call the dyad the
line itself, others the form of line”’ (this is consistent with Ross, Met. 2.203 and van der
Wielen 144ff). For, to paraphrase the justification, with the dyad, for example, the form is
nothing else than the dyad itself, whereas with the line—the ideal line—one must dis-
tinguish between its form—the dyad—and a second constituent (the short-and-long). The
equation of line and dyad belongs to Xenocrates, and we may attribute the other version
of the theory of ideal numbers to Plato. To be sure, the argumentation of the μὲν γάρ
sentence seems to be directed against Xenocrates, and would thus have to go back to a
third Platonist, who may have defended Plato’s proposal. There were too many partners
to the discussion, too many varieties of the doctrine, for our reconstruction to attain
absolute certainty.
46 Cf. Becker in Fs. Gadamer; W. Heisenberg, Naturwissenschaften 45 (1958) 227-234.
47 Tim. 35aft; cf. ch. V 1, below.
48 So Ross, PTI 210, De Vogel, Mnemosyne 1949, 304f, Saffrey passim. No reliance
may be placed, admittedly, in the fact that Simpl. De ai. 28.7ff and Philop. De an. 75.34f
refer to On the Good. Cf. Cherniss, Plato 119 n. 77.
49 So Cherniss, Plato 565ff, Riddle 14f; Theiler, Arist. 94.—Kucharski sought to prove
that Pythagorean doctrine was in the background here, relying on the late evidence of
Aétius and Theo Smyrnaeus (below, ch. I 3, n. 109) and on the alleged “ caractére artificiel
et naif” of the doctrine (Arch. 43)—-though it does not seem to have been too primitive
for Xenocrates. First he disposes of the theory of ideas (Tétr. 36f, 47ff), though it is un-
mistakably presupposed (of μὲν yap ἀριθμοὶ τὰ εἴδη αὐτὰ... ἐλέγοντο), then later (Arch.
33f, 39f) scems willing to consider whether this “point neuralgique”” may after all
show Platonic influence. But according to Aristotle’s explicit testimony the theory of
ideas is Platonic and not Pythagorcan. Cf. also Saffrey 8ff.
δ0 Cherniss, Plato 568f; Ross, PTT 210; De Vogel, Mnemosyne 1949, 304f; Saffrey 7;
Theiler, Arist. 93. The same form of citation is found at Phys. 194236.
In gg2a1off, Aristotle presents the derivation of line, plane, and solid in the first
person. (On such expressions in the early books of the Metaphysics sce Jaeger,
Arist. 171, 188, Eng. ed.) In addition, Alexander refers for this (Met, 117.24) tothe Περὶ
φιλοσοφίας.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
(ἔτι δὲ καὶ ἄλλως) adds still another version of the doctrine, without
citation of the source. The identification of the One with mind (νοῦς)
is attested for Xenocrates, as well as the series knowledge-opinion-
sensation,” and he knows the derivation-series line-plane-solid (above,
n. 39). But then Xenocrates’ definition of soul® is introduced as
something new, a conclusion of “‘some’’; therefore what precedes
can not refer to Xenocrates,*4 and it is a tempting conclusion that what
follows the citation of the Timaeus and its interpretation in the work
On Philosophy is a reference to “unwritten doctrines.”
The close connection of all this with Plato is shown by a remarkable
passage in the Laws (8044) :*8
The origin of each thing takes place . . . when a first principle, taking
on increment |the line], passes into its second transformation [the
plane] and from this to its neighbor [the solid], and having made
three transformations makes perception possible to those who
perceive it.
Plato is dealing with the relation of the soul to the physical world, the
priority of soul over matter. Thus there are present in Plato both the
series line-plane-solid and the equation of the last stage with percep-
tion,>? though to be sure there is nothing about the application of
numbers.
Finally, we must take account of Theophrastus’ statement that
‘most’’ of those who posit the One and the Indefinite Dyad as first
principles only carry the derivation to a certain point:
¢
52 Fragments 15 and 5. Cf. Heinze 2ff, Cherniss, Plato 57of. The series νοῦς -ἐπιστήμη--
δόξα -αἴσθησις is familiar to Aristotle himself (De an. 428a4, Met. 1074b35).
53 Frr, 60-65 Heinze.
54 Set forth by De Vogel, Mnemosyne 1949, 304. Cherniss, Plato $73f, emphasizes the
οὕτως, which he refers to γνωριστικόν, as a connection with what precedes, and he does
not cite the “‘some” in the middle. But the word κινητικόν introduces a new topic; and,
besides, οὕτως is to be referred to the main clause (Theiler, Arist. 94). Theiler (ibid.)
acknowledges that “Aristotle would have had to express himself differently, if he had
already cited Xenocrates’’; but then he separates off the last clause as an addendum, and
still, because of 1090b2off, attributes what precedes to Kenocrates. Yet he recognizes
that there is a difference from the definition of soul, which was “perhaps only formulated
later.” So we have both the early and the late Xenocrates—too complicated a solution.
55 Cf. Phys. 209br1ff.
56 The connection was already noted in England’s commentary. Cf. Cornford, PlParm
198; Kucharski Tétr. 71ff. Also Pl. Leg. 819¢7: ἀριθμοί... μήκη, πλάτη, βάθη.
57 The series ἐπιστήμη-δόξα- αἴσθησις is also found at Parnt. 155d. Cf. 142a, 164b, Tim.
28a, 37bc, §2a.
1. The Platonic Theory of Ideal Numbers
for having generated numbers and planes and solids, they practically
omit the rest, except for a brief mention, just enough to make clear
that some arc from the Indefinite Dyad . . . and some from numbers
and the One, like soul...
Spcusippus treats the matter similarly, he says, but Xenocrates differ-
cntly (Met. 6a25ff).5® Again we have an ontology, separated from
Speusippus and Xenocrates, in which the series number-line-plane-
solid is present alongside the connection of numbers and soul. Thus for
the pioneering exposition of the theory of ideal numbers, with its
iuplications for the physical world and the soul, we find ourselves led
back again and again to Plato, and apparently to the lecture On the Good.
The commentators on Aristotle, from Alexander of Aphrodisias on,
are unanimous that the doctrines developed in On the Good were
Pythagorean,®® and Aristotle also says that in his theory of the first
principles Plato “mostly” followed the Pythagoreans, though he did
have “something of his own” to add.6° The seventh Letter, in the
passape on these doctrines (338c), makes reference to Archytas: from
lus circle, it is suggested, Dionysius might have had knowledge of
doctrine which Plato, himself, had not imparted to him.*! But, even
if this relationship may be regarded as certain, still the question becomes
wll the more urgent what it may have been that suggested this course to
Plato. That he did not take over someone else’s system unchanged may
le assumed from the start, and not only because of its psychological
tnprobability. When Plato went to Magna Graecia for the first time,
at the age of nearly forty, his intellectual attitudes must have been fairly
well established. The main background for Plato’s ontology is clearly
Hleatte dialectics: it is not by chance that Parmenides and the “‘Eleatic
attanger’” play a leading part in the late dialogues; the One and the
(rood had been equated already by Euclides of Megara, a doctrine
“ Vhese follow remarks on Hestiaeus and Plato. L. Taran (AJP 87 [1966] 471) draws
Hie conclusion ¢hat Plato cannot therefore have been referred to earlier. But μέχρι τῶν
εἰρημένων (μι 4) looks back τὸ a23ff.
™ Alex. Met. §§.20, Simpl. Phys. 151.13ff, De au. 28.7ff, Philop. De an. 75.34ff. Whether
Armatle hiuself spoke of Pythagoreans in his Περὶ τἀγαθοῦ is, however, doubtful;
whee Alexander quotes exactly (46.35) there is mention only of Plato.
ἢ Βα 106. C4. ch. 12.
Hhate only mentions Archytas here to explain how Dionysius might have arrived
ab these rapaxotopara, Prank assumes (243.0) that Plato himselfspoke of Pythagoreanism.
(toch. 4. below.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
Plato is echoing in the Protagoras.*® Above all, Aristotle clearly dis-
tinguishes between Platonism and Pythagoreanism; starting from his
statements, it is possible to judge the originality of Plato, as well as to
find out what Pythagorean philosophy was like before Plato.
It is true that Aristotle’s reports on the history of philosophy harbor
the numerous and serious sources of error which Cherniss, in particular,
indefatigably exposed. But in default of other sources, there is nothing
else to do than—with due caution—to follow Aristotle's hints. Perhaps
we may hope that those distortions, and the faulty perspective, which is
present to an equal degree in the reports on Platonists and on Pythag-
oreans, will prove negligible in assessing the differences between the two.
2. THE PHILOSOPHY OF THE PYTHAGOREANS ACCORDING TO
ARISTOTLE
Aristotle does not present his reports on the Pythagoreans as an
impartial historian of philosophy, but always in the context of his own
exposition, which has purposes of its own. The thing that lends the
confrontation with the Pythagoreans its special immediacy for Aristotle
is their connection with the Academic teaching on first principles,
which he criticizes vigorously while at the same time making it his own
point of departure.t And, in the organization of his inquiries about
first principles (ἀρχαί) in the first two books of the Metaphysics, the
Pythagoreans are not only treated separately, among the “pre-
Socratics’’ (985b23ff), but are even more often treated in comparison
with Plato.? Their doctrines are set forth in detail in the discussion of
the difficulties in the Academic number theory (in books M and N).3
The latter is the real target of Aristotle's polemic; Pythagorean
material may be cited for its own sake, in order to achieve complete-
ness, or it may be used to win a point from the Platonists.4 There are
further detailed reports in the Physics, in the discussion of the concept
ἄπειρον (202b30ff). Isolated reports on items of scientific theory are
62 D.L. 2.106; Cic. Acad, 2.129. Pl. Prot. 329cd (O. Gigon in Phyllobolia: Festschrift
P. von der Mill (Basle, 1947] 100). Cf. Kramer so5ff.
1 Thus Aristotle finds the pattern of ἕν and ὑπεροχὴ καὶ ἔλλειψις in nearly “all” of the
“early” thinkers (Phys. 189b8ff).
2 987a20ff, y89b20ff; cf. 990a27ff, go6a4ff.
3 Elsewhere in the Metaphysics, there is only one allusion each in Z (1036618), 7 (1053b12),
and A (1072b31).
"Ch 1o8gbkE: ὁ δὲ τῶν {Πυθαγορείων τρόπος τῇ μὲν ἐλάττονας ἔχει δυσχερείας...
Logoazo: of μὲν οὖν Πυθαγόρειοι κατὰ μὲν τὸ τοιοῦτον οὐϊενὶ ἔνοχοί εἰσιν.
᾿
κα
2. The Pythagoreans according to Aristotle
also found in other works; the famous cosmic system, with its moving
earth and harmony of the spheres, is discussed in the book On the
Heavens. |
In addition, Aristotle devoted two special books to the exposition
and criticism of Pythagorean doctrines, and also wrote on Pythagoras.
He himself alludes once (Met. 986a12) to his “more exact”’ discussions.
Plutarch, Alexander of Aphrodisias, Aelian, and especially Iamblichus
have preserved important material from these books, which supple-
ments the reports of the didactic treatises.®
It has been emphasized repeatedly that Aristotle, in his extant works,
consistently speaks of Πυθαγόρειοι, not of Pythagoras,® and this seems
° The list in D.L. 5.25 (on which see P. Moraux, Les listes anciennes des ouvrages
d' Aristote [Louvain, 1951] 243ff, I. During, Aristotle in the Ancient Biographical Tradition
|(séteborg, 1957] 67ff) names one book each Πρὸς τοὺς Πυθαγορείους and Περὶ τῶν
ΠΠυθαγορείων; the catalogue of Ptolemaeus (Moraux 289ff, During 208ff) gives two books
(nu the Doctrines of Pythagoras. The first book is cited by Aétius 1.18.6 (Arist. fr. 201 Rose),
the second by Alex. Met. 75.16f, Simpl. Cael. 392.16ff (Arist. frr. 202, 205 Rose); obvi-
ously the two books had been combined into one (Moraux 301). The Hesychius catalogue
las only one book, Περὶ τῶν Πυθαγορείων (Moraux 198). The various titles under which
the work is cited are given by Wilpert (Hermes 1940, 373 n. 3), who has also considerably
auiyginented the material to be drawn from Alexander, beyond Rose’s collection. For
corrections relating to Rose’s collection, which was taken over in essentials by Ross, see
below, ch. II 4, ἢ. 4. Further material is to be found in Iamblichus; see below, n. 112,
ch. U4, n. 5, ch. II 5, nn. 14-17. Plutarch uses the work (Gell. 4.11.11ff = Arist. fr. 194
{axe}, sometimes without naming it (De E 388b-c; Quaest. Rom. 288c, with fr. 199 and
ay Rose; De Is. et Os. 364a, with fr. 196 Rose; Numa 8, with fr. 191 Rose).—The
aithmological fragment published by Delatte (Litt. 167ff) from a codex of the 16th
century is derived in essentials from Alex. Met. 38.8ff, with added details. It shares with
the Alexander MSS (at 38.21) the corruption καρπῶν (instead of καιρῶν, found in Ascl.
Mct. 36.6, who is dependent on Alexander), with the principal MSS (at 38.19) ἐτῶν (the
parallel tradition for Alex., MSS L and F, and Ascl. has μηνῶν) and (at 38.16) the non-
νον] στερεός (the correct τετράγωνος in L and F). If the fragment agrees with corrup-
tions which came after the split in the transmission of the text of Alexander, it is dependent
ou alate form of that text and not an independent witness to a source antedating Alex., as
Helatte thinks.—Rose (De Aristotelis librorum ordine et auctoritate [Berlin, 1854] 81ff,
At istoteles pseudepigraphus |Leipzig, 1863] 193f) denied the authenticity of the book on the
I'ythagoreans, as indeed of all the lost works of Aristotle. (Thus Rohde, Q 103 n. I and
juss, also speaks of “pseudo-Aristotle.”) This rejection still finds champions today,
capecially because of the collection of miracle tales the book contained (Wehrli, Herakleides
+t, anthe other side, Zeller I 396, n. 1; Corssen, RhM 1912, 36ff). There can be no ques-
tion of a nco-Pythagorean forgery, as the citations in Bolus-Apollonius Hist. mir. 6 give
vtetminus ad quem. The question whether Aristotle himself or one of his pupils collected
the material is irrelevant for its reliability (Diels, AGP 1890, 467 n. 39, ΗΟΙΚ 22, Delatte,
bint 42}
"οἱ καλούμενοι Πυθαγόρειοι, 985623, 98gb29, Mete. 345014, Cael. 284b7; τῶν
Tradimaw τινες καὶ καλουμένων Πυθαγορείων, Mete. 342b30; of περὶ τὴν ᾿Ιταλίαν, καλούμενοι
δὲ {Πηαγύρειοι, Cael. 293020; ᾿Ιταλικοί, Met. οϑ7αγι. At 986a20f the chronological
“ἀπο! about Pythagoras is an interpolation (Zeller 1 $97, n. 2; Ross, Met. 1152; Jaeger,
Paideia 1063, 0.72 [Eng. trp and his note on the passage in the OCT Met.). It may stem
fom ban. PP rog; itis not found in Alexander's paraphrase or in the principal MS, but
Aw αν το it (Met. 39.25) On MM τὐῆλατα and tr, 207 Rose, see below, ch. 13, mn. 164).
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
also to have been the case in the lost works, as far as philosophical
doctrines are concerned.” Often he adds the word καλούμενοι, and the
extent to which this expression, “τῆς so-called Pythagoreans,”’ shows
some kind of reservation has been vigorously debated.* There lies
in the Greek expression about the same kind of nuance that is ex-
pressed nowadays by quotation marks. But the studious avoidance of
Pythagoras’ name is intentional.
The doctrine of the Pythagorcans usually appears as an undifferen-
tiated unity; or in any case there is no sure foothold for the modern
attempts to discern various stages in its development.® To be sure,
“others of this same group”’ (ἕτεροι τῶν αὐτῶν τούτων) are named as
originators of the “table of opposites,’’!° differences on specific points
3
are mentioned twice," and several times there is a restrictive ““some’
(τινές) 13 Yet there is never a distinction between “‘early’’ and “late”
Pythagoreans.
For the comparison of Plato and the Pythagoreans we may begin
with the first book of the Metaphysics.3 Plato’s doctrine is at first con-
nected closely with the Pythagoreans: “in most respects following
them, but possessing also some features of its own which set it apart
7 Citation of the title as Περὶ τῆς Πυθαγόρου φιλοσοφίας, Aét. 1.18.6, proves nothing
(in spite of Guthrie I 200): the later tradition often has “‘ Pythagoras” instead of “‘ Pytha-
goreans”; cf. D.L. 3.8, with Arist. Met. 987a32ff; Olympiod. In Meteor. §0.9ff, on
342b30, and n. 5 above.
8 Frank uses the adjective sogenannt in the title of his book, as does Bollinger (and see
her pp. 15f). Cf. the expression τὰ καλούμενα ᾿Ορφικά, Arist. Gen. an. 734a19, De an.
410b27, with Philoponus’ comment ἐπειδὴ μὴ δοκεῖ ᾿Ορφέως εἶναι τὰ ἔπη. Contra,
Cherniss (Pres. 384) calls attention to οὗ καλούμενοι γεωργοί (Pol. 1290b40), and speaks of
‘‘designations in the currently recognized sense” (Gnomon 1959, 37£); cf. ΖΜ 354. Von
Fritz (AnnMath 1945, 249.38) suggests that there may bea certain reserve intended, because
it was unusual for a philosophical school to be named in this way after its founder; but cf,
᾿Αναξαγόρειοι, Pl. (ταί. 409b, Dialex. 6.8 (DK II 414.13), “Hpakdeirecot, Pl. Tht. 179¢,
D.L. 9.6. Zeller’s observation is important, that there are names of political factions in
-εἰοι (I 446 n. τ; Minar 21). E.g., Διώνειοι, Pl. Ep. 7.334c; Διαγόρειοι, Hell. Oxy. 10;
Κυλώνειοι, lam. VP 248. Cf. F. Poland, Geschichte des griechischen Vereinswesens (1909) 73ff.
Since the formations in -e.o. are regular only for s-stems (Bin ‘HpaxAnein, but ayopa—
ἀγοραῖος), the form Πυθαγόρειοι must depend on earlier examples, but probably in the
realm of the ἑταιρίαι.
® Cornford himself admits that it is difficult to divide Aristotle’s reports between the
two “radically opposed” schools he reconstructs.
10 986222. Cf. the end of this chapter.
1 Mete. 345a14ff (Milky Way), De an. 404016ff (motes in the air and souls). The theory
of comets in Mete. 342b30ff and the system of the τὸ heavenly bodies are mutually
exclusive (below ch. IV 1).
12 σινές is even used in relation to the theory of numbers in Cael. 300817 (glossed over by
Zeller | 4sof, 451 0. 2), and in relation to particular questions Cael, 2g3b21, Mete. 342b30,
Sens. 446016. In Aristotle, τινές can also refer to a single author, as Pol, φόρα (Hide. 3.20),
Cen, an. 7 «οι 4 (Edt. 2.93).
On the problems of the analysis of y87a2gfF, in connection: with to78b 2th, see
Cherniss, Plate 1.77 n. τοῦ, and 1git.
2. The Pythagoreans according to Aristotle
from the philosophy of the Italians’’ (987a30f). To be sure—and we
must be mindful of this qualification—this relates exclusively to the
question of what principles (dpyai) the early thinkers proposed, and
how they arc related to the Aristotelian classification. Plato and the
Pythagoreans both accepted numbers as the principles (987b24)—
number not as the number of other assumed objects, but as an inde-
pendent entity, οὐσία. In this sense Pythagoreans and Plato are regularly
mentioned in conjunction, and are set over against all other thinkers
carlier than Aristotle, and in this conjunction Aristotle regularly
attacks them. At the same time, though, Aristotle emphasizes the
difference that, while Plato separates the numbers, as ideas, from the
sensible world and even sets between them the mathematical realm as
a realm of its own, for the Pythagoreans things “are’’ numbers, they
‘consist of’ numbers.!> Aristotle is puzzled at this: the Pythagoreans
introduced principles, he says, that would have been quite suited to
lead beyond the merely perceptible into the higher realms of Being,
but then they never speak of anything but what is perceptible, and
“squander” (καταναλίσκουσιν) their principles on this world of ours,
as though there did not exist anything but what is perceptible, what
the sky encloses (989b29ff). What differentiates Plato from the Pythag-
oreans is “separation’’ (χωρισμός), the “introduction of the Forms”
(987b31), which Aristotle traces back to the dialectic developed by
Socrates.1®
If the numbers are identical with things, they are space- and time-
bound. Aristotle actually speaks of their origin, in a cosmogonic
sense,!? and says that they are extended, that their units possess mag-
nitude.!® It is naturally against this thesis that the principal logical
and physical objections are directed® and Zeller wanted to ex-
clude the report about “extended units” completely, as Aristotelian
"1 τὸ μέντοι ye ἕν οὐσίαν εἶναι, καὶ μὴ ἕτερόν τι ὃν λέγεσθαι ἕν... 987b22 (cf.996a6f),
ΙΟΟΙ ΔΟΒ 1053b11ff, Phys. 2ο344 (also on the ἄπειρον).
ι5 O87b28 (cf. τὸν ἀριθμὸν τὰ ὄντα, 1083b17); 1090a22; ἀριθμοὺς τὸν ὅλον οὐρανὸν,
ὡδόλδε (cf. 986a3); τὰ σώματα ἐξ ἀριθμῶν, 1083b11ff (cf. Cael. 300a14ff; 986b6; 990421;
"ΟΝ 002, 16, 18), ἐξ ἀριθμῶν τὰ ὄντα, 1090423, 32. But we also find ἐξ ἐκείνων (sc. σωμάτων)
ὄντων τῶν ἀριθμῶν, 1083b18; τοὺς ἀριθμοὺς ev τοῖς αἰσθητοῖς, 1080b1. The lack of
χωρισμός, 1080b16, τοϑθιο, 1090a23, Phiys. 203a6ff; on this point the Pythagoreans are
to be praised (above, n. 4)—Aristotle himself argues against χωρισμός
τ᾽ ro78b3off; O. Gigon, “Die Sokratesdoxographie bei Aristoteles,’ MH τό (1959)
7a 212.
'7 Below, αν 3S.
IW τὰς μονάδαν ἔχειν μέγεθος, tovobry; cf b33, τοῦ ις, ggoara.
" Logical objections: mathematical number has no magnitude, 1o80br2ff Cael.
yoou RL; physical: the force of gravity (ggoargt, bogoazat, Cael. yooarg) and the pheno-
menon of movement (ygoak) both remain unexplained.
Ἢ
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
interpretation.?° But at least there can have been nothing to refute such
an Aristotelian interpretation. The Pythagoreans did not differentiate
between number and corporeality, between corporeal and incorporeal
being.?! Like all the pre-Socratics, these Pythagoreans take everything
that exists in the same way, as something material. For every Platonist
this exposition of Aristotle’s is an excruciating annoyance, which even
the superficiality of the commentators of late antiquity could not cause
them to overlook. Syrianus felt this most keenly, and devised a truly
modern remedy:
Aristotle has no reliable or adequate objection to the dpyai of the
Pythagoreans. For the most part, if I am to speak the truth frankly,
he does not even hit them, but launches his objections against
hypotheses he has invented himself.”
Proclus too, and others, are certain that only a misunderstanding or a
malicious distortion could be to blame.?° But what is an embarrassment
for the Platonist is treasure-trove for the historian: here we have a piece
of Pythagorean doctrine that was not subsumed into Platonism.
The differences extend further. ““Elements’’ of the numbers are,
according to Pythagorean doctrine, the ‘““even”’ and the “odd”; the
“odd” is at the same time “limit,” the “even” is “unlimited.” In the
pair limit-unlimited we have a primeval cosmic opposition lying
behind the number which is the world.*4 To Greek linguistic feeling,
“limit” is the positive principle ;*5 it is conceived at the same time as
20 Zeller I 486. On ‘‘number atomism”’ and Ecphantus, see below, nn. 66, 74.
21 So also Raven in KR 247 n. 1. This makes it improbable that the word ἀσώματος
was coined by the Pythagoreans, as H. Gomperz tried to show (Hermes 67 [1932] 155-167).
(On Philolaus B22, see below, ch. III 2, ἢ. 45.) We may speak of the “immanence of
number,” but it is impossible to equate Pythagorean doctrine with the conception of
Aristotle, whose theory of abstraction was possible only after the development of the
theory of ideas.
22 Syrian. Met. 80.20ff; cf. 83.12 ff.
23 Procl. In Tim. I 16.29; cf. Ascl. Met. 34.15ff. Simplicius’ style of interpretation is
fairer, though likewise permeated by reverence for Pythagoreanism: πῇ καλῶς ἐκείνων
λεγόντων ὁ ᾿Αριστοτέλης πρὸς τὸ φαινόμενον ἀντεῖπε τοῦ λόγου (Cael. 386.8, on Arist.
Cael. 284b6). Cf. Simpl. Phys. 453.10ff, 652.6.
24 τοῦ δὲ ἀριθμοῦ τὰ στοιχεῖα τό TE ἄρτιον καὶ TO περιττόν, 986a18. At 1004b31 the
pairs odd-even and limit-unlimited are separated, but this may be merely ἃ slip. The point
is enumeration of oppositions of any kind. (Cherniss, Pres. 47, 186-188, assumes that for
the second pair Platonists are included, but there is no evidence for this.) Limit-unlimited
comes first at ggoa8 and 986a23 (the table of opposites); only limit-unlimited at 1091a17
(cf. EN 1106b28, fr. 47); only odd-cven at Phys. 203a10ff; the explicit equation of even
and unlimited at Phys. 2o3a1off. On 987a16§ cf. below, n. 38.
35 °The basic importance of this opposition of limit and unlimited was set forth by
Heidel (AGP 1y01), in opposition to Zeller. He leaves the question open whether the
opposition of odd and even was originally equal or subordinate to it (τοῦ), but if one
considers that it must have been from a Platonic perspective chat Aristotle brought the
2. The Pythagoreans according to Aristotle
masculine, the “‘unlimited”’ as feminine, and correspondingly the odd
number is also masculine, and the even feminine.*® Aristotle gives a
complicated explanation of the correspondence of odd and limit, even
and unlimited, at the basis of which lies the representation of numbers
by arrangements of pebbles.?? Since Burnet the significance of this
number aspect of Pythagorean doctrine into the foreground, one will be still more con-
vinced of the importance of the cosmological limit-unlimited. Cf. D. N. Levin, Ethical
Implications of the πέρας-ἄπειρον Dichotomy as Seen Particularly in the Works of Aeschylus,
!Diss. Harvard, 1957 (summary in HSCP 63 [1958] 519-522).
26 This is why the number 5 (=2-+ 3) is ‘‘ marriage” (γάμος). Aristotle alludes to this at
Met. 1078b23 and fr. 203=Alex. Met. 39.8ff. Also Plut. De E 388a-c, Quaest. Rom. 288c.
( below, n. 31.
27 Phys. 203a13: περιτιθεμένων yap τῶν γνωμόνων περὶ TO ἕν καὶ χωρὶς ὁτὲ μὲν ἄλλο
ἀεὶ γίγνεσθαι τὸ εἶδος, ὁτὲ δὲ ἕν. On this see Stob. 1 prooem. 10(=DK 58Β28), Alex-
ander ap. Simpl. Phys. 457.12ff, Philop. Phys. 394.1ff, Themist. Phys. 80.13ff; with more
detail Theo 31.15ff, Nicomachus Th. ar. 9.16f€ (with the textual emendations of Becker
ZwU 24f), Nicom. Ar. 1.9.4, 2.17, Iam. In Nic. 73.15ff. The gnomon spoken of here is
ἈΠ instrument for measuring right angles, like a carpenter’s square. The result of its
application as described is shown in the adjacent figure. Cf. Burnet, EGP 103f; Ross,
odd even
0 00 0 0° 0000
eee 0 eee eo
0 0 @ 0 © 00 @ 0
@oeo @® @0oeo
Met. 1 148£; Ross, Phys. $42ff; Raven, PyEl 130f, 188ff; KR 243ff; Becker, Grdl. 34ff;
(suthrie I, 242ff. Aristotle’s expression is not clear: καὶ χωρίς is an old crux interpretum;
but che inclusion of the opposition square-rectangle in the table of opposites favors the
usual interpretation; with the odd numbers squares result, with the even rectangles. For
other, more complicated explanations, see Taylor, CR 40 (1926) 149ff, and M. Timpanaro
Cardini, Physis 3 (1961) 105ff.—Simpl. Phys. 455.20ff (=DK 58B28; cf. Philop. Phys.
sXy.11, 391.25, Themist. Phys. 80.9f) attributes a different interpretation to certain
ἐζηγηταί: the even number can be halved, but halving continues to infinity κατὰ τὴν
διχοτομίαν. Precisely because of the apparent irrationality of this, some have sought to
tind here an ancient doctrine, like W. A. Heidel, AGP 1901, 395f (followed by Burnet,
ECP 288f; Ross, Met. I 149), and with variations Raven in PyEl 193 and KR 244{.
What the exegetes mean, however, can be deduced from Porphyry ap. Simpl. Phys.
443.25ff: the number 2 is the principle of division which proceeds to infinity, and only to
this extent is there a connection between even numbers and the principle of infinity.
These considerations derive from the thought of the Indefinite Dyad, and are therefore
not carly Pythagorean.—Raven, PyEl 130ff, and Kucharski (“Les principes des Pytha-
yoticiens et la dyade de Platon,” Archives de philos. 22 [1959] 175-191, 385-431) have
πο, not incorrectly, that in the gnomon procedure One and Dyad are given equal
tank as principles of the odd and even numbers respectively: but the further thesis, that
the Dyad in this function could have been called Indefinite, and that therefore all the
later tradition was right in designating the concept of the Indefinite Dyad as Pythagorean,
ἐπ rest on the formulations cited from Nicomachus, Theo, and Iamblichus. All the
detailed accounts ascribe to Pythagoras the unequivocally Platonic concept of the
Indefinite Dyad, καθ᾽ ὑπερβολὴν καὶ ὄλλειψιν (see below, ch. 1 3), but Aristotle speaks in
this passage of ἄπειρον, not of ἀόριστον, and nowhere else makes any mention of an
προσίῃς cole played by “twoness” in Pythagoreanism. When Philip says (105), “the
Wlustration is Aristotle's,” he is forgetting that it is reported in indirect discourse.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
allusion to the graphic basis of Pythagorean number speculation has
been recognized;?8 it is not the cipher or numeral (like ‘‘s5’’) which
serves as pictorial representation of a number, but the shape of an area
—the sort of thing we are familiar with from dice or dominoes. Léon
Brunschvicg introduced the apt allusion to the constellations, where,
too, in each case a limited number of points of light defines an object:
Orion, the Bear, the Lion, etc.28 One need not believe, however, that
any one pebble arrangement was the sole point of departure for the
basic identification of limit and odd. Even numbers can of course be
halved, until one comes to odd numbers, and then the halving process
s “at its limit.’” More general considerations also enter in: like the
positive evaluation of limit, the habit of prizing odd numbers has for
long been deeply rooted in popular fecling.° Plutarch, what is more,
ina passage where at least for part of the way he is following Aristotle,
gives a perspicuous explanation, also from the viewpoint of the pebble
figure, of the masculine character of the odd number and the feminine
of the even. The even number, he says, has at its middle an empty
space, capable of reception, whereas the odd number has a middle
member with procreative power.®! This direct symbolism must be
regarded as old, and not only because of its attestation; it has a con-
nection, at least subliminally, with the general Greek association of
masculinity with the word zepaivew.®
From the two primal principles rises the One; from it, number.
The similarity of this to the Platonic system, to the derivation of
28 EGP tor1ff. Zeller (I 483ff) had denied the spatial character of numbers in Pytha-
goreanism. Cf. above, n. 20.
29 Tes étapes de la philosophie mathématique, 3rd ed. (Paris, 1929) 33: “‘si non l’origine,
du moins V illustration saisissante de la doctrine pythagoricienne.”’ |
30 On this, see below, ch. VI 4.
31 Plut. De E 388a-c. The first part, the explanation of the ἕν ἀρτιοπέριττον, corre-
sponds to Arist. frr. 199 and 203 (Alex. Met. 40.18ff), Quaest. Rom. 288c; Stob. 1 proem
10; cf. Aristox. fr. 23. The One also has a mid-position in odd numbers according to the
Platonic doctrine (Arist. Met. 1083b28).
even Oo oO
odd @ @ e@ marriage Φ Ὁ @® O @
32 T.¢., in the sexual sense of wepaiveey.—Cherniss (Pres. 17 n. 68, 38) conjectures that
Aristotle first thought of the equality of even and unlimited under the influence of the
Platonic conception of the Unlimited Dyad. But not only the connections shown here,
but also the pebble procedure, which Aristotle surely did not invent, would count against
this. —Movas masculine and δυάς feminine are also found in Xenocrates (fr. 1.4}, ma imyth-
ical form: of expression, Cf. also Philolaus f, 200.
Mokout7: τοῦ δὲ ἀριθμοῦ στοιχεῖα τό τε ἄρτιον καὶ τὸ περιττόν, τούτων δὲ τὸ μὲν
πεπερισμένον τὸ δὲ ἄπειρον, τὸ δὲ ὃν ἐξ ἀμφοτέρων εἶναι τούτων {καὶ γὰρ ἄρτιον εἶναι
καὶ περιττόν), τὸν δ᾽ ἀριθμὸν ἐκ τοῦ dvdy,
2. The Pythagoreans according to Aristotle
number from the Once and the Indefinite Dyad, is obvious. Aristotle,
too, sets Unlimited alongside Indefinite Dyad, but at the same time
emphasizes what is “peculiar”’ (ἴδιον) in Plato’s formulation: “the
fact that in place of the Unlimited, treated as singular, he posits a dyad,
and derives the Unlimited from the great-and-small—this is peculiar
to him” (987b25).4 The term Indefinite Dyad is thus by no means
Pythagorean in origin; but the difference goes still deeper. The
Unlimited, unitary and undifferentiated, was according to Pythag-
orean conceptions “outside the heaven’’ and penetrated the world
by being “breathed in” by the heaven to separate natural things
(φύσεις) from one another, being “enclosed and partitioned off”
(ἐναπολαμβανόμενον) in the limited.2> Obviously the concept of empty
space, as well as that of air, is present here. In any case the Unlimited
of the Pythagoreans is principle and constituent of the cosmos at the
same time, whereas the Unlimited Dyad of the Platonists is conceived
not cosmically but ontologically, even logically. It is not in any place,
but is a transcendental principle, in which everything in the realm of
experience takes part, and even, in a different way, in the realm of
Idcas;37 it belongs to the theory of ideas, just as the cosmic Unlimited
of the Pythagoreans fits consistently into a world without a conception
of incorporeal being.
Insofar as ἄπειρον and ἀόριστος δυάς are comparable, the One falls
naturally into the company of πέρας, and Aristotle occasionally speaks
34 Similarly, Phys. 203a15.
ὅδ Phys. 203a6: ...€lvar τὸ ἔξω τοῦ οὐρανοῦ τὸ ἄπειρον... τὸ ἄπειρον εἶναι τὸ
ἄρτιον τοῦτο γὰρ ἐναπολαμβανόμενον καὶ ὑπὸ τοῦ περιττοῦ περαινόμενον παρέχειν
τοῖς οὖσι τὴν ἀπειρίαν... 213b22: εἶναι δ᾽ ἔφασαν καὶ οἱ Πυθαγόρειοι κενόν, καὶ
ἐπεισιέναι αὐτὸ τῷ οὐρανῷ ἐκ τοῦ ἀπείρου ἵ πνεύματος ὡς ἀναπνέοντι καὶ τὸ κενόν, ὃ
διορίζει τὰς φύσεις, ὡς ὄντος τοῦ κενοῦ χωρισμοῦ τινὸς τῶν ἐφεξῆς καὶ [τῆς] διορίσεως"
καὶ τοῦτ᾽ εἶναι πρῶτον ἐν τοῖς ἀριθμοῖς" τὸ γὰρ κενὸν διορίζειν τὴν φύσιν αὐτῶν...
lor the obviously corrupt wording Raven proposes(KR no. 315), with Diels, πνεῦμά τε;
von Fritz (RE) suggests ὡς πνεῦμά τι ἀναπνέοντι [καὶ] τὸ κενόν. Cf. fr. 201 Rose:
row μὲν οὐρανὸν εἶναι ἕνα, ἐπεισάγεσθαι δ᾽ ἐκ τοῦ ἀπείρου χρόνον τε καὶ πνοὴν καὶ τὸ κενὸν
ἡ διορίζει ἑκάστων τὰς χώρας ἀεί. See also Cael. 279a11ff, and Cherniss, Pres. 214f. On
the relation of this to Xenophanes, see below, ch. III 3 n. 17 (with context).
“ "I'he difference between space and air was established by Anaxagoras (A69) and
iimpedocles (B100). To draw chronological conclusions from this fact, leading to an early
ating, of these Pythagorean doctrines (Raven, PyEl 28ff, 44ff) is, however, dangerous;
the “breath”? of mythical cosmogonies may have survived merely in the comparison of
votd and breath; cf. above, n. 35. The logical difficulty, that the Unlimited becomes a
hinting agent, only becomes acute because of Aristotle’s terminology.
"7 ddmaros ix quite seldom attested before Plato, but ἄπειρον as περιέχον is familiar to
the pre-Socratics from the time of Anaximander (A1o, 14; cf Anaxagoras B 1, 2). Later
Mhatonists find i¢ important to note that the term “unlimited” is to be understood not in
the sense of spatial extension or corporeal multiplicity, but ontologically (Plut. Quaest.
ce, 7Ige-d), as “indefinite,” not “infinite.”
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
as though the two were to be identificd.®* Yet precisely because this is
so obvious—even absolutely necessary—from the point of view of
Platonism, one must believe the definite testimony of Aristotle that
in the Pythagorean view the One had its origin from Limit and
Unlimited together: τὸ δ᾽ ἕν ἐξ ἀμφοτέρων εἶναι τούτων (986aI9). For
the Platonist this is no less a vexation than the lack of incorporeality;
μᾶλλον δ᾽ ἀμφότερα ἐκ τοῦ ἑνός are the words with which Asclepius
impatiently interrupts his paraphrase of this passage of the Metaphysics
(38.25). The One has a share in each of the opposite forces; it is “even”’
and “odd”’ at the same time, ἀρτιοπέριττον 339 it is perfectly in keeping
that it should be, as late sources say, bisexual, ἀρσενόθηλυ.40 This One
comes into being and develops further—it is nothing else than the
world before its further evolution. The “‘first principles,’ Limit and
Unlimited are, then, what was there before the world came into being.
To the Pythagoreans, number philosophy is cosmogony: κοσμοποιοῦσιν
(τορτ418). The further development of number is also cosmogonic:
They say clearly that when the One had been constructed—whether
of planes or surface or seed or something they cannot express, then
immediately the nearest part of the Unlimited began “to be drawn
and limited by the Limit.’”41
The One becomes a Two as the Unlimited penetrates it. Here is one
38 Aristotle characterizes the One (ἕν) and the Unlimited (ἄπειρον) as οὐσίαι (above,
n. 14), and to this extent sets them side by side (987417); he calls the One “‘element and
principle” at 1080b31,and only names the Unlimited as “principle” at 988426 (differently,
986bsff). At 987a16, if one strikes out καὶ τὸ ἕν, with MS E, the identification of ἕν and
πέρας is perfect, for then the pairs τὸ πεπερασμένον Kai τὸ ἄπειρον at als and αὐτὸ τὸ
ἄπειρον καὶ αὐτὸ τὸ ἕν at a17 correspond; but, since Alex. Met. 47.11 read καὶ τὸ ἕν (whereas
the equivalence of One and πέρας would seem natural to any later, Platonic-minded
reader), we must keep the full text. Aristotle is enumerating unsystematically: both the
primary pair of opposites, and the One, and number in general, are οὐσίαι for Pytha-
goreans; in his repetition, Aristotle leaves out the πέρας that goes with the ἕν, following,
consciously or not, Platonic ways of thinking (cf. Cherniss, Pres. 45 n. 175, 224ff). In
Aristotle’s own philosophy, πέρας is οὐσία and περαινόμενον is ὕλη (Cael. 293b13); and
he can, under the influence of Platonic terminology, even ascribe to Empedocles the
doctrine that τὸ ἕν is a στοιχεῖον (Gen. corr. 315a23).—One cannot derive from the table
of opposites the equation one=limit=good (as KR 241). This would imply that a
rectangle has crooked sides, since “rectangular” is in the same column as “crooked.”
39 Frr. 199 and 203 Rose (the latter, Alex. Met. 40.18ff); Plut. De ΠῚ 388a—c; “‘for it is
both even and odd,” Arist. Met. 986a20. Raven (KR 317) concludes from the table of
opposites that “one” was originally regarded as odd; against this, cf. above, ἢ. 38.
40 Nicom. Th. ar. 4.1, 4.17ff, Macrob. Somn, Sc. 1.6.7.
41 davepws yap λέγουσιν ὡς τοῦ ἑνὸς συσταθέντος, εἴτ᾽ ἐξ ἐπιπέδων εἴτ᾽ ἐκ χροιᾶς
εἴτ᾽ ἐκ σπέρματος εἴτ᾽ ἐξ ὧν ἀποροῦσιν εἰπεῖν, εὐθὺς τὰ ἔγγιστα τοῦ ἀπείρου ὅτι εἵλκετο
καὶ ἐπεραίνετο ὑπὸ τοῦ πέρατος (τόφτα 0). The ὅτε cannot be deleted; it shows ἃ
citation is τὸ follow, and probably separates paraphrase from: quotation. Ch below
ch. War, g2; Met. ro8ob2of, Phys, δε Δ ΠῚ with fr. 201, and above, αν 14: also ΒΟ 14:
γεννῶσί re yap τὸν οὐρανόν,
2. The Pythagoreans according to Aristotle
of the most widespread cosmogonic themes, “the separation of
Heaven and Earth”’:
ὡς οὐρανός τε γαῖά τ᾽ ἦν μορφὴ μία.
ἐπεὶ δ᾽ ἐχωρίσθησαν ἀλλήλων Biya...”
The process was modified by the Pythagoreans, with their ideas of the
carth in motion and the central fire. But in these very ideas is apparent
a complete equivalence of the things separated: the “Hearth”’ of the
universe and its fiery envelope, “Zeus’s castle’’4? and Olympus.“ This
separation has happened in the past: when Aristotle quotes exactly,
verbs in past tenses suddenly appear. ,
The growth of the cosmos is described, in the manner of the pre-
Socratics, as the growth of a living being, and embryological concepts
form part of the background.*® The One begins to breathe,** and, as
the breath flows in, it assumes a more complicated structure. Similarly,
the Hippocratic book De natura pueri*’ teaches that the embryo
develops as the seed, in the womb, takes in air, and is divided and
articulated by it: “Each of these [bodily structures] is articulated by
the breath; for as they are aerated by it they separate according to their
natural affinities.” In Aristotle’s account the origin of the One from
seed is at least mentioned asa possibility.4® Thus it becomes obvious that
the ancient idea of macrocosm and microcosm is at work in Pythag-
orcan teaching.*® It is not a matter of an ontology featuring the
42 Eur. fr. 484; on the idea, W. Staudacher, Die Trennung von Himmel und Erde, Diss.
Tubingen, 1942.
43 Zavos πύργος, Διὸς φυλακή, Διὸς θρόνος Arist. fr. 204- Simpl. Cael. §12.12ff; Zavos
φυλακή Procl. In Eucl. 90.17.
44 Philolaus A16; cf. Parmenides fr. 11.2, Emp. fr. 44, Hebd. 2, below, ch. III 2. In a
similar way, Anaximander already has fire envelop the earth “like bark,” and then be
"broken away” (Ato; Kahn $7f).
41 Cf. H. C. Baldry, ‘‘Embryological Analogies in Pre-Socratic Cosmogony,” CQ 26
(1932) 27-34.
40 ἕλκειν is a technical term for breathing: Hippoc. Nat. puer. 12, Nat. hom. §, Flat. 3,
(Jar. 6, Diog. Apollon. A31, Emp. A74.
47 VII 498 L. The connection with Pythagorean doctrine was shown by Olerud, $3ff.
(if) VIL 488 L.: ἡ γονὴ ὑμενοῦται φυσωμένη. .. Philolaus also has a similar doctrine, that
a living creature after its birth “immediately” draws in breath and cools itself (A22; cf.
Cruthrie 1 278f).
4" As emphasized by Cornford, CQ 1923, 10 ἡ. 1; PlParm 19; cf. KR 251, Guthrie I
AGH.
On this, Kranz, NGG 1938 and ΑΒΕ 1955; Olerud passim. To the context of these
anthropomorphic” conceptions of the cosmos belong the speculations about the “right”
and ‘left’? stde of the world (Arist. Cael. 284b6ff, 285b25ff, fr. 200, fr. 205). We need
not here go into problems of detail, such as the contradiction that Alexander found in
Avistotle’s statements (fr. 205); see Zeller 1 447. ἡ. 1, Cherniss, Pres. 186 n. 178.
1 Caillandre’s study is unhelpful (La droite ct la gauche dans les podmes homeériques en
concordance avec la doctrine pythagoricicnne et avec la tradition celtique |Paris, 1944]). On
λα μη ἡ and ‘left’ in general, G. ER. Lloyd, ΜΙΝ 82 (1962) 96-66; in Oriental
coumology, Κα Sethe NGG, μηνοῖν, Ki, ty22, 1.7 242.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
derivation of number-in-itself; their number theory is cosmogony.*°
Here Pythagorean doctrine is quite in line with pre-Socratic tradi-
tions. The cosmogony of Leucippus is closcly related: he too has two
kinds of entity at the “beginning’’; the many atoms and the empty
space between that makes possible their separation. The first step
toward organization in the cosmic whirl is that a “membrane” or
“caul” (dprjv—here too a term used in embryology) detaches itself
from the nucleus, and the cosmos grows as matter flows in from the
exterior.*1 But beyond the pre-Socratic horizon a mythical one also
becomes visible: the separation of Heaven and Earth, and above all the
double nature of the primally existent. For whatcan only be explained
artificially, in arithmetical terms, is familiar to the mythologist—the
primal being that is bisexual.*
Perhaps a quite specific mythical cosmogony forms the background
of the Pythagorean number theory. There are striking similarities of
detail in the Orphic cosmogony which in the romance of Pseudo-
Clement is given by Apion as an example of pagan theology. The
problems of transmission are exceedingly complicated,® but the basis
is unquestionably a hexameter poem ascribed to Orpheus. Allegorical
interpretation of Orphic poems, from a philosophical point of view,
goes back at least to the fourth century 8.c.,as the papyrus from Dher-
veni has proven;* so it is quite possible that in the tradition of philo-
sophical exegesis ancient material has been preserved. In specific
50 Cherniss (Pres. 30ff, 44ff, 224f, 387f) tries to show that the Pythagoreans had no
doctrine of the origin of number, and that Aristotle only produces this impression by his
projection of Platonic ways of thinking. To establish this he must, at Met. 1og1a13ff,
make a radical separation of the cosmic One (‘‘the universe itself”) and the “‘numerical
unit”: ‘‘Aristotle is confusing ... the cosmogony with the number-theory”’ (p. 39).
But Aristotle says unequivocally that the Pythagoreans knew only one kind of number,
the cosmic (990a21), that is, that they thought of number theory as cosmogony, of
cosmogony as the development of arithmetic.
51 Leucippus AI=D.L. 9.32: τούτου δ᾽ οἷον ὑμένα ἀφίστασθαι... αὐτόν τε πάλιν
τὸν περιέχοντα οἷον ὑμένα αὔξεσθαι κατὰ τὴν ἐπέγκρισιν (ἐπέκρυσιν MSS) τῶν ἔξωθεν
σωμάτων. (On the text and interpretation see J. Kerschensteiner, Hermes 87 [1959] 441-448,
esp. p. 446.) The agreement is still greater when the question is not of an abstract πέρας
but of a plurality of wepaivovra (below, ch. III 2).
δὲ Cf. Olerud, pp. 130-136; H. Baumann, Das doppelte Geschlecht, (Berlin, 1955);
M. Delcourt, Hermaphrodite (Paris, 19§8).
53 More detailed treatment of the following, A GA 14 (1968) 107ff. See Clem. Hom.
6.38, Rec. 10.17ff; 30ff; Kern, Orph. frag. 5-56. On the Clement narrative in general, sec
the introductions to the new editions by B. Rehm (Homilies [Berlin, 19§3]) and B. Rehm
and Εν Paschke (Rekognitionen | Berlin, 1965|); also Rehin’s article in RAC 3 [19$7] 1978).
Eusebius knew the Apion dialogues (Hist. ecel. 3.38.5).
4 Arch. delt. τὸ (1964) 171f. Interpretation of Orphic material in Cleanthes and Chrysip-
pus, SUP TL go, go7, 1078.
2. The Pythagoreans according to Aristotle
details, this ‘‘Orphic” text gives an impression of antiquity,” and it
parallels to a surprising extent the first stages of the Pythagorean
number theory. In the beginning was an abyss, a ‘boundless sea,” a
limitless chaos. In it there came to be, by and by, and for no particular
reason, a “bubble,” which began to grow and become firmer. It
sucked in the surrounding πνεῦμα, its “skin” became hard, and soon
there floated on the sea of boundlessness a glittering sphere: the world
egg. In this there deyeloped a living creature, like the sphere in shape,
winged, bisexual. It broke the egg and “appeared” in radiant brilli-
ance: Phanes! Then the two halves of the broken shell fitted themselves
together “harmoniously,” while Phanes took position at the utmost
boundaries of the heavens, a secret, spiritual light; and from the
‘“procreative”’ content of the egg arose the realms of the world.**
Orphism and Pythagoreanism were almost inextricably intertwined
in the fifth century (cf. ch. II 2), so that it is understandable that, within
the pre-Socratic domain, Pythagorean doctrine developed as a trans-
posed version of Orphic cosmogony. We have only a few crumbs of
knowledge about further details of their cosmogony: ten heavenly
bodies came into being, which circle about the central fire. There were
10 because 10 is the “perfect” number;5” and it seems also to be
significant that, counting from the periphery, the sun is in seventh
ὅδ Apion’s source is not to be identified with the cosmogony of “Hieronymus and
Ε 1ellanicus” (Orph. frag. 54, 5786), where Kern placed the Apion testimonies. The ‘‘ Rhap-
sodic Cosmogony” is closer, if πνεῦμα is a synonym for αἰθήρ (Orph. frag. 70). There,
lhowever, Chaos and Aether are “begotten” by Chronos (Orph. frag. 66), whereas,
according to Apion, Chaos had always been there (Hom. 6.3.1, 6.4.1, Rec. 10.30.3), and
Chronos plays no active role. But Chronos is also missing in the Orphic cosmogony
known to Aristotle (Met. 1071b27) and Eudemus (fr. 150), and in the parody of Aristo-
phanes (Av. 693ff). The bisexual Phanes seems to be reflected in the myth of Plato’s
Symposium (189d; cf. K. Ziegler, NJb 1913, 529ff; RE s.v. Orphische Dichtung 1361f).
‘he breathing in of a πνεῦμα seems already to have a part in the Egyptian story of the
world egg (S. Morenz, in Aus Antike und Orient: Festschr. W. Schubart, 1950, pp. qf;
cf, also Quellen des alten Orients: Die Schépfungsmythen, Ziirich 1964, p. 90) and in Mochus,
lGsrHist 784F4; Gen. 1.2.
68 Hom. 6.4.1ff (Orph. frag. 55): ἀπείρου τινὸς βυθοῦ ἀεὶ ῥέοντος... 2: συνέβη ποτὲ...
εὐτάκτως ῥυῆναι... ὥσπερ ἴλιγγα καὶ μεῖξαι τὰς οὐσίας... ὥσπερ... πομφόλυξ....
καὶ τὸ περικείμενον πνεῦμα ἐπισπάσασθαι... 3: ὑπὸ τοῦ περιειληφότος θειώδους
δ Arist. Met. gkoaktt; cf below, ch. IV 1.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
place.58 If we see number here in a quite different function—not as a
spatially extended figure but as an ordinal—yet the statement that the
whole universe is ἁρμονία καὶ ἀριθμός leads off in a still different direc-
tion :59 number is “harmony” not only in a general way, as that which
transcends the opposition of Limit and Unlimited, but even more so
on the basis of the recognition that all musical intervals are determined
by numerical ratios.6° This music then moves out into the cosmic
realm, in the idea of the “harmony of the spheres’’; a contributing
factor in this development is to be found in scientific insights, in a
modest knowledge of the periodicity of the movements of the planets,
and in hypotheses about their distances and speeds.* All the same, the
« . « . £6 33 .
special position of these items of “knowledge,” as compared with
“speculations,” does not seem to have been prasped. Rather, we find
even more bizarre statements: 1 is mind (νοῦς), 2 opinion, 3 the number
of the Whole, 4 (or 9) “justice,” 5 “marriage,” 7 “right time” (καιρός),
and τὸ is “perfect.”’®? Aristotle tries to understand all this as attempts
at scientific definition, though he also complains of “ superficiality.’
It is even more amazing, though also consistent, that these abstract
concepts (or perhaps more correctly: these powers) each takes its
specific position in the cosmos—for the cosmos itself does consist of
numbers.®4
From our perspective, we may distinguish at least four quite different
functions of number on which the Pythagorean doctrines are built:
number as the symbol of certain concepts or powers of ordering;
number as the designator of order, position, or rank; number as
determiner of spatial extent (the pebble figures); and finally number
as ratio and mathematical formula, as natural law. Aristotle already was
58 Fr. 203= Alex. Met. 38.20ff.
59 o86a2f, 21.
80 985b31: τῶν ἁρμονιῶν ἐν ἀριθμοῖς ὁρῶντες τὰ πάθη καὶ τοὺς λόγους.
Cf. below, ch. V.
61 Cf. below, ch. IV 4.
62 Below, ch. VI 4.
83 987220, 1078b21ff, EN 1132b2iff, MM 1182aI1.
64 ggoa8ff; on this, Alex. Met. 75.15, with the all-too-brief remark that in his second
book On the Pythagoreans Aristotle dealt with the arrangement of the numbers in the
heaven. Alexander says here that the One had the mid-position, and thus must have been
counting from the middle out; in 38.21ff= Arist. fr. 203, however, where we learn that
the sun is 7, the enumeration goes in the other direction. One would not expect purely
arbitrary improvisations in Alexander, so that the irrational ambiguity must be Pytha-
gorean. For surprising combinations see Aristotle’s collection, presented with an air of
mockery, in 1093a13ff. Here there is Platonic material intermingled, but the expression
οὐλομέλεια τοῦ οὐρανοῦ is most probably Pythagorean. (οὐλομέλεια is an Tonic word,
attested most often in the Hippocratic writings: DK [458 n.; attributed to “Orpheus” at
Th. ar. 48.6.)
2. The Pythagoreans according to Aristotle
complaining, in his day, that very different things were being equated
here; no wonder that historians of philosophy, with so many different
points of departure, have arrived at quite disparate results. Some
interpret the Pythagorean number theory as a radical materialization of
number and find it a kind of atomism, while others see in it a philos-
ophy of mathematical form, an idealism closely akin, if not identical
to, the Platonic theory of ideas.
The “number atomism”’ interpretation goes back to Cornford.® In
his account of Pythagorean doctrine, Aristotle speaks of a plurality of
extended monads,® and he often alludes to the definition of the point
as a “‘monad having position.’’6° If we interpret this as a comprehensive
key idea, to be taken along with the pebble figures, the “star pictures”
(constellations), and the procedure of Eurytus, who would determine
the “number” of man or horse by making an outline picture with
pebbles*®—the result is the thesis that the Pythagoreans understood the
materialized point as a kind of atom. They thought of all bodies as
consisting of such point-atoms, and therefore things “are’’ numbers in
the most literal sense; that is, they are the number of atom-point-units
which they at any given moment contain. Does not Aristotle himself
say that the Atomists “in a way’ claim that things ‘‘are numbers or
composed of numbers” (Cael. 30348)?
Still, though Aristotle’s refutation sometimes presupposes an
atomistic view,’® Cornford’s theory cannot claim to give the final
answer. Every atomic theory operates with invisibly small “units,’’ so
that every visible body consists of an incalculably large number of
them; and in fact Cornford speaks throughout of “indefinite plural-
ity.”7! But is an incalculably large number still a ““number’’? Would
‘6
δ ggoa22ff: one should distinguish, he says, between the number “out of which the
world is composed,” the “ magnitudes being composed”’ (26), on one hand, and numbers
as causes of properties like opinion or opportunity. (Cf. 986a16, 987a22ff, 1036b17, and
10g2b8ff; and 1093a1ff, where the opponents in mind are mainly the Platonists.)
66 Cornford, CQ; PlParm 56ff. Cherniss, especially, is in agreement (Pres. 39f, 387); he
thinks this interpretation gives him a basis from which to criticize Aristotle’s reports.
Laven expresses doubts, PyEl 67ff, KR 247 (though on pp. 248 and 249 objects are said
to be “aggregations of unit-point-atoms.’’)
8? ro80bry9, 1083b15; cf. above, nn. 15, 18. 1084b26ff is aimed at the Platonists.
68 Below, ch. I 3, nn. 86-87.
% Arist. Met. 1og2broff; Theophr. Met. 6argff, after Archytas. Further reports by
ps.-Alex. Met. 827.9 (DK 45) are of dubious authenticity.
7 Ac Met. 1083b8ff, Aristotle asserts, as refutation of Pythagorean views, that there are
no “indivisible magnitudes.” But, on his own principles, Cherniss ought not to have
assuuned from this (Pres, 20) that the doctrine attacked had precisely the content which
the refutation presupposes. |
1 OQ 1922, 137, PlParm 60.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
it not be a better conclusion—on the sorites principle—that a unit
amounts just about to nothing? If this doctrine is supposed to be early
Pythagorean, we must reject the majority of the testimonies of
Aristotle. According to all the sources, the fundamental thing for the
Pythagoreans was the numbers from 1 to 10, not myriads; the impor-
tant thing is form, not statistics, and this includes the method of Eurytus,
who looked for the significant points, the specific criteria of the shape.”
Nor can Pythagorean atomism be deduced from the arguments of
Zeno;”3 and how could anyone put more trust in conjectures about
putative opponents of Zeno, whom he does not himself name, than in
what Aristotle ascribes explicitly to the Pythagoreans? If Ecphantus
was indeed “the first”? to attempt an atomistic interpretation of the
number theory, this is an attempt to modernize the theory, rather than
a revelation of its original significance.”
Another line of thought mentioned by Aristotle is almost always
taken as Pythagorean: the “limit” (πέρας) is the essential nature
(οὐσία) of a thing, so that a plane surface is more οὐσία than the physical
object, a line more than a plane, a point—and thus a unit—still more
than a line. Since this argument is contrasted by Aristotle, in one
passage, to that of Plato, Speusippus, and Kenocrates (though Xeno-
crates is not named), it has been thought that by a kind of process of
elimination we may decide for the Pythagoreans.” In addition, a
doxographical report names μαθηματικοί as adherents to the limit
doctrine, and people have assumed without hesitation that “mathe-
maticians’’ must mean Pythagoreans.”¢
72 So Raven, PyEl 104f, KR 313f.
73 Below, ch. III 3.
74 DK 51.2; G. Vlastos, Gnomon 25 (1953) 32f; below, ch. IV 3, ἢ. 17.
75 The passages are 1002a4ff, 1028b16ff, 1o6ob12ff, 1ogobsff, and perhaps Cael. 298b33.
Diels included this material (DK 58B23-24), as did Kirk and Raven (320, as “pre-
Parmenidean,” and 405, in the chapter on Philolaus and Eurytus; the phrase “by a process
of elimination,” p. 316). See also Ross, Met. Π 162, Kucharski, Tétr. 27f, Guthrie I 259.
Zeller saw the right answer (I 484 n. 1: ‘‘Plato”; to be sure, the problem of the “ points”
arises again; cf. above, ch. I 1, ἢ. 20), as did Cherniss (Pres. 40ff ; Plato 132ff) and Saffrey
27%.
τὸ Ps. Galen Phil. hist. 18 (Dox. 611.2, 23; 613.1), Sext. Emp. PH 3.32, Math. 9.364,
367ff. For the assumption that references to μαθηματικοί are to Pythagoreans, cf. Frank,
n. 174, and Logos 9 (1920-1921) 246 n. 1; Rougicr 23, 64, etc. Examination of the data
reveals that μαθηματικοί are advocates of astronomical theories in Aét. 2.15.5, 2.16.2,
2.16.7, 2.30.7, 2.31.2, 2.29.6; astrological, 5.18.6. The statement in Aét. 2.31.2 on the
distance of the moon from the earth is decisive: it is the view of Aristarchus of Samos
(Dox. 63). At Aét. 4.14.3 Pythagoreans and μαθηματικοί are named as sponsors of the
theory of mirror images, and with this belongs the theory of vision in 4.13.9 (the same
comparison with the stretching out of the hand): The astronomer Hipparchus is named
here: ἔνιοι δὲ καὶ Πυθαγόραν τῇ δόξῃ ταύτῃ συνεπιγράφουσιν dre δὴ βεβαιωτὴν τῶν
2. The Pythagoreans according to Aristotle
To be sure, the underlying consideration, that there can be a surface
without a body, but no body without surface, line, and point??—this
“conceptual experiment” is fundamental for the system of derivation
found in Aristotle’s On the Good.’® But for the Pythagoreans, who
know of nothing else than what is sensually perceptible, there can
scarcely be a “limit” without a body (ἄνευ σώματος, 10026). In fact
the limit doctrine appears once where the point of view is that of
χωρισμός, 79 and this shows it cannot be Pythagorean. There is a passage
in which the thought seems to be that the limit is immanent: To a
mention of the limit doctrine, Aristotle adds the question, whether
things of this kind exist inside or (as Plato, Speusippus, and Xenocrates
have it) outside the realm of the sensible.8° Yet some Platonists main-
tained the immanence of mathematicals.*! For the Pythagoreans even
the primary One is three-dimensional, or corporeal. The reduction of
the physical world in the schema of body-surface-line-(point) belongs
to the Platonists, not to the Pythagoreans, who knew only the one
world of the sensible.
Now it is true that a much-cited passage of Aristotle does seem to
confirm an idealistic interpretation of the Pythagorean number
doctrine, the famous sentence in which the difference between Plato
and the Pythagoreans seems to shrink down to nothing but an altera-
tion of terminology: τὴν δὲ μέθεξιν τοὔνομα μετέβαλεν [Πλάτων7᾽ of μὲν
γὰρ Πυθαγόρειοι μιμήσει τὰ ὄντα φασὶν εἶναι τῶν ἀριθμῶν, Πλάτων δὲ
μεθέξει, τοὔνομα μεταβαλών (987bi0ff). If it was only that “he changed
the word,” then, it is suggested here, the relation of number and
things in Pythagorean doctrine was of just the same kind as that of
μαθημάτων... That is, even in antiquity, we have a conjecture based on the postulate
that mathematics is Pythagorean. Ps.-Plutarch (Aét. 4.14.3) left out the μαθηματικοί and
speaks only of Pythagoreans; this is how the tradition burgeons. What we must hold to
is that the μαθηματικοί of the doxographical tradition are professional mathematicians
and astronomers of the Hellenistic age. (This is also true for Por. ap. Stob. 1.49.61, in
spite of Delatte, Litt. 126. Cf. also Cic. Div. 2.91 Sext. Emp. Math. 10.174 Plut. Prim.
frig. 9$3a.)
7? 100226: καὶ τὰ μὲν (sc. ἐπιφάνεια, γραμμή, στιγμή) ἄνευ σώματος ἐνδέχεσθαι δοκεῖ
εἶναι, τὸ δὲ σῶμα ἄνευ τούτων ἀδύνατον.
* Arist. fr. 28= Alex. Met. ες. 22ft (cf. above, ch. 1 1): τῶν Se σωμάτων πρῶτα τὰ ἐπίπεδα
εἶναι --τὰ yap ἁπλούστερά τε καὶ μὴ a λει πρῶτα τῇ φύσει---, ἐπιπέδων δὲ
γραμμαὶ. - γραμμῶν δὲ στιγμαὶ.
Ἢ oben διὰ τί οὖν χωριστὰ ἔσται at b13.
“ 1o28b18: ἔτι παρὰ τὰ αἰσθητὰ οἱ μὲν οὐκ οἴονται εἶναι οὐδὲν τοιοῦτον, οἱ δὲ πλείω
καὶ μᾶλλον ὄντα aidia, ὥσπερ Πλάτων, Σ᾽ πεύσιππος....
κι ρυδαγῆ (Gimmanence of mathematicals, separation of forms); cf. 1076a38ff. Other
Phatonists, not precisely identifiable, are inchided along with Plato, Speusippus, and
Xenocrates at 108§a13, 1O87b1 4th, voksb2ae.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
idea and thing in Plato; and all attempts to prove a Pythagorean
origin for the theory of ideas make use of this argument.®
The difference between this and the reports elsewhere that for the
Pythagoreans things “are’’ numbers, seems insuperable. The concept
of development might seem to offer a solution of the dilemma, but its
sponsors offer opposite formulae. If the identification of thing and
number seems to Burnet primitive and therefore old,®* to Cornford
the idea of “imitation’’ seems “mystical,” and therefore original.*4
Others have decided to eliminate one of the two versions as a faulty
tradition, but, here again, sometimes one and sometimes the other is
rejected. 88 |
Amazingly enough, Aristotle himself does not seem to have noticed
the discrepancy; the phrase ἀριθμοὺς εἶναι... αὐτὰ τὰ πράγματα stands
just a little further on (b28). It is quite unlikely that Aristotle was
following different sources; could the contradiction be only apparent?
Evidently Aristotle knew the Pythagoreans had used the word μίμησις.
He surely must have intended to belittle Plato’s originality in the
doctrines which Aristotle himself disliked. Thus the thought that this
Pythagorean μίμησις was the same as μέθεξις, and therefore implied a
theory of ideas, will be seen to be interpretation. Aristotle does give
some clues as to how imitation can be understood without a theory of
ideas. The word “‘immanence” is no help (above, ἢ. 21), but he does
say that the Pythagoreans supposed they saw ὁμοιώματα between things
and numbers,®’ and it is only natural to express such similarity or
correspondence by the word μίμησις. Similarly, in the Hippocratic
writings the relation of microcosm and macrocosm becomes a matter
82 Cornford, PrSap 46: “The Platonic theory of ideas is described by Aristotle as a
variety of Pythagoreanism . . .” For a Pythagorean theory of ideas, cf. Burnet, EGP 308f,
ThPL 151f, Plato’s Phaedo (Oxford, 1911), introd. pp. xliii-xlvi; Taylor, VarSocr 178ff,
Delatte, Pol. 108f.
83 EGP goff, 307ff; now add Philip 73.
84 CQ 1922, 143f; 1923, 5.
80 Since the notion of imitation scemed more fruitful philosophically, it is accepted by
Gilbert, AGP 1939, 39f; by Jacger, Paideia I 163 (Eng. tr.), for whom, in the other reports,
“he is no doubt making the mistake of translating into material terms their theoretical
identification of numberness and existence’; by Ross, PTI 217 (only “a primitive
savage" could simply identify thing and number); and by Maddalena, 265 ἢ. 24. Taking
the opposite position, Cherniss, Pres. 386ff, says the imitation-statement came from a
source who wished to contest Plato’s originality, perhaps Aristoxcnus (392: cf. Aristox.
fr. 23). Similarly, Frank thinks of Speusippus (256).
4 Sce the attempts at compromise by Zeller (1 4stf) and Rey (356ff), who ingeniously
suggests that things “are” numbers when one considers their basic nature, but “imitate”
numbers when one considers their properties. Also see Raven ΠΥ ΠΟΙ, Guthrie 1 230f,
OXshaz;, ἀφωμοιῶσθαι, by, ἀφομοιῶν (πε γεν), royabrz. These references are
adduced by Zeller (1 4.410} and also discussed by Mondaolto (7M 354).
2. The Pythagoreans according to Aristotle
of “imitation,” but—and here is the surprising fact—this imitation
may be turned either way. One may just as well say that the human
body “imitates” the cosmos as that the parts of the cosmos “‘imitate”’
human organs.®® In the same way, either the arts imitate nature or
nature imitates the arts.6* Imitation is a two-sided correspondence,
which makes it possible to interpret separate things following the same
pattern, but without implying differences of rank or a relationship of
ontological priority. Indeed, it scarcely makes a difference whether one
says that stone “imitates” bones, or that earth “is’’ flesh.°° When we
place the Pythagorean theory in this pre-Socratic context, Aristotle's
statement about “imitation” falls into place with the rest. Nothing
more is meant than the correspondence of cosmos and number, in the
sense that one explains and illuminates the other.*! In post-Platonic
thought one can scarcely speak of imitation without assuming that it
implies a gradation of kinds of Being, especially since Plato often
characterizes the relation of sensible object and Idea as μίμησις.» To
this extent the surprising report of Aristotle is comprehensible. For the
historical placement of the Pythagorean number doctrine, the result is
more a confirmation than a correction, to say nothing of an earth-
shaking contradiction.
Again and again it becomes clear that the Pythagorean doctrine
cannot be expressed in Aristotle’s terminology. Their numbers are
“mathematical” and yet, in view of their spatial, concrete nature, they
are not.*4 They “seem” to be conceived as matter (ὕλη) and yet they
are something like form (efSos).® They are, in themselves, being
88 For the first-named expression see Hippoc. Vict. 1.10, for the second Hebd. 6.1, lines
13, 23 (imitatio). For Plato, to be sure, there is a basic difference: οὐ yap γῆ γυναῖκα
μεμίμηται... ἀλλὰ γυνὴ γῆν (Menex. 2384).
89 Hippoc. Vict. τ τ
90 Hebd. 6.1. (If we may trust the Latin translations, the predicate was missing in the
Cireek, in this case.) “‘Imitate”’ and “‘be” are related in the same way as simile and meta-
phor: the latter is morphologically earlier, but the two can stand side by side.
1 Cf. Phys. 213b26: καὶ τοῦτ᾽ εἶναι πρῶτον ev τοῖς apiOuots—‘ first’? because most
clearly and therefore most essential: 987222: ᾧ πρώτῳ ὑπάρξειεν ὁ λεχθεὶς ὅρος, τοῦτ᾽
εἶναι τὴν οὐσίαν τοῦ πράγματος; Phys. 203a12f: σημεῖον δ᾽ εἶναι τούτου τὸ συμβαῖνον ἐπὶ
τῶν ἀριθμῶν. 1o2ob4f is not specifically Pythagorean, but contemporary mathematics
(Kucl. 7, defs. 17~18).
%2 "Tim. 38a, 39c, 48e-f, etc.
% Cherniss (Pres. 2866) finds a third version of the relation of thing and number in the
expression, ‘the elements of numbers are the elements of all things” (986ar; cf. 985b25,
and for the formulation 987b18; contra, Guthrie 1 229f).
"4 The Pythagorean number is “ mathematical” (1o80b16) but “not unitary” (ovadtxds,
rokobty, bz2); still one can say, “an arithmetical number is unitary” (1083b16); so that
the corporeal number of the Pythagoreans is not mathematical (1083b12),
QkOb6: ἐοίκασι δ' ὡς ἐν ὕλης εἴδει τά στοιχεῖα τάττειν; YROAIG: καὶ ὡς ὕλην τοῖς
οὖσι καὶ ὡς πάθη re καὶ ἕξειν, cf also above, 1. ὁ 4.
I, PLATONIC AND PYTHAGOREAN NUMBER THEORY
(οὐσία), and yet are not quite 50.386 They cannot be expressed in the
Aristotelian framework of the four principles, or in the categories of
form and matter; great as the temptation has been, both for Aristotle
and for modern scholars, to understand the opposition of Limit and
- Unlimited as identical to that of form and matter, the explicit state-
ments that the One partakes of both Limit and Unlimited, and that
number is a kind of material (ὕλη), stand in the way. Missing are the
impact of the theory of ideas and the dialectic, the classification of
Being into stages of differing reality (οὐσία), the reduction of the sen-
sible world to immaterial principles. Neither the system of Aristotle
nor the conceptual framework developed by the Academy forms any
part of the background of these Pythagorean doctrines; rather, they
obstruct our access to them and impede our understanding of them.
When one puts these observations alongside the traces of cosmogonic
myths which dominate the apparently abstract pattern of the genesis
of the ““numbers,”’ there can remain no doubt. What Aristotle presents
as the philosophy of the Pythagoreans is truly pre-Socratic, unaffected
by the achievements of Socratic-Platonic dialectic, and not to be
measured by their standards. Of yap πρότεροι διαλεκτικῆς οὐ μετεῖχον
(Arist. Met. 987b32).
Thus the thesis of Frank and Howald, that the whole number
philosophy of the “‘so-called Pythagoreans” was developed only
within the precincts of the Old Academy, is refuted from within the
theory itself.®’ Indeed, Aristotle says, repeatedly and unambiguously,
that the Pythagoreans are to be dated earlier than Plato.°® Elsewhere,
to be sure, Aristotle's chronological indications are imprecise enough
to make one think that he was not himself quite clear about the order
of events. The surprising anonymity of the doctrines is of a piece
with this chronological indefiniteness. The only personal name that
certainly belongs with this philosophy is ““Pythagoras’’; but Aristotle
purposely avoids it. The “Pythagoreans’’ belong among the “later”’
pre-Socratics.!°
8 ἄπειρον, πέρας, ἕν, ἀριθμός as οὐσία, above, ἢ. 14; nevertheless, σαφῶς μὲν οὐ διήρθρωται
παρ᾽ ἐκείνων, 986b5f; cf. Phys. 204033f.
Frank 248: “Thus there is nothing in the whole of Aristotle’s account of the Pytha-
gorean philosophy which could not be derived from Speusippus or similar Platonists.”’
Cf, his next pages; similarly Howald, fs. Sudhoff 7off, with amplification, Bollinger 4off.
O87a29tF, b32, 1053b12, 10o78b20.
Their relation to the Atomists is expressed very vaguely at 98$b23, in the words
ἐν δὲ τούτοις καὶ πρὸ τούτων (ct Alex. Met. 347.6). At toz8bar the Pythagoreans are
dated carlier than Democritus.
M At roadartl Platonists and Pythagoreans are mentioned as “those who came later
and seemed wiser” (οἱ δ᾽ ὕστεροι καὶ συφώτεροι,, .. δόξαντες). This counts against Philip's
thesis chat che whole doctrine goes back to Pythagoras himself (446).
2. The Pythagoreans according to Aristotle
As to the sources to which Aristotle owes his knowledge, there are
two different clues. The story about Eurytus is expressly referred by
Theophrastus to an account given by Archytas,1%™ and here we obvi-
ously have oral tradition transmitted through Academic connections
in Magna Graecia; Speusippus and Xenocrates had been in Sicily with
Plato.1 In the second place, Aristotle occasionally plays off the
Pythagorean doctrines against the Academy in such a way as to make
the conclusion unavoidable that he is using written sources without
Academic coloring.1° Therefore he must have had at least one original
Pythagorean document.
The scanty indications about Archytas given by Aristotle and his
pupil Eudemus reveal an advanced stage of Pythagorean doctrine, not
far from Platonism. Archytas proposes definitions whose purpose is
differentiation between matter and form,!°4 while Pythagorean defini-
tions had been “‘superficial,’’!°° and their number theory made no
distinction between form and matter. Archytas designated the “irregu-
lar,” or the “‘indefinite,’”’ as cause of motion,!°* whereas in the Pythag-
orean number theory the cause of motion remained unclear (99048).
If we add that the procedure of Eurytus, too, seems to be an attempt
at a more comprehensive systematization of the number theory,!°” and
that Eurytus is to be dated earlier than Archytas, the terminus ante
quem recedes still further back into the fifth century. But Aristoxenus
and later authors mention one name in conjunction with that of
Eurytus—namely, Philolaus (below, ch. HI).
When one attempts to understand the Pythagorean nnmber theory
in the context of fifth-century pre-Socratic thought, it is impossi-
ble not to become aware, along with the mythical-cosmogonical
ll Met. 6a19: 6 περ "Apytras ποτ᾽ ἔφη...
2 Plut. Dion 22 (the third Sicilian journey). Cf. Merlan, Philologus 103 (1959) 203;
‘Vimaeus FGrHist 566F158.
103 Esp. 1091a13ff.
104 1043a19ff=DK 47A22. To be sure, Aristotle is ‘‘systematizing,” as his remarks
show; but in the interest of a genuine definition Archytas had already gone beyond the
Pythagoreans’ symbolic numbers.
05 Above, n. 63. .
6 Eudemus fr. 60 Wehrli: Πλάτων δὲ τὸ μέγα καὶ τὸ μικρὸν καὶ τὸ μὴ ὃν καὶ τὸ ἀνώμαλον
. τὴν κίνησιν λέγει... βέλτιον γὰρ αἴτιον λέγειν ταῦτα ὥσπερ ᾿Αρχύτας.... τὸ δὲ
ἀόριστον, φησί, καλῶς ἐπὶ τὴν κίνησιν οἱ Πυθαγόρειοι καὶ ὁ Πλάτων ἐπιφέρουσιν (οὐ γὰρ
δὴ ἄλλος γε οὐδεὶς περὶ αὐτῆς εἴρηκεν)... Thissecond sentence, introduced by Simplicius
with the words ‘and a little further on,” obviously refers to the preceding, and the
Keneral expression “ Pythagoreans” stands for Archytas (DK §81B32 removes the context.)
Aristotle gives Plato’s doctrine, without naming him, and adding his own criticism, at
Phys. λοι Ὁ © Met, ro66arolf. In subject matter, Eudemus closely follows his teacher.
7 Hurytus pets ironical praise trom ‘Theophrastus because he goes further than anyone
else into absurdity!
4
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
background and their proximity to Leucippus, ofa certain kinship with
the Eleatics.1°8 In terms of Aristotle’s criticism, the Eleatics and the
Pythagoreans find themselves near neighbours. Both schools, and they
alone among the pre-Socratics, pressed forward to the brink of
immaterial being, but in neither case were they able to make any real
use of this advance. For the Eleatics, too, though “transcending sense
perception” (Gen. corr. 325a13), “still conceived of ‘existing things’
as being only sensible things” (Met. 1oroarff; cf. Cael. 298b21), just
as the Pythagoreans squander their principles on the sensible
world (989b29ff), while at the same time mistreating it in the
process: “They seem to be speaking of another heaven and other
bodies than the perceptible” (1090a34), and the theses of the Eleatics
about our world actually seem to “border on madness” (Gen. corr.
325aI9).
A striking fact about the examples that Aristotle gives for the
Pythagorean equation of numbers and things is that itis never a question
of the relation of individual thing and individual number—aside from
the isolated fooleries of Eurytus—but of the correspondence of a
plurality of things to the system of numbers, and in particular the
correspondence of alterations in things to alterations in the number
scrics. As Aristotle puts it, πάθη of things correspond to πάθη of
numbers.!°® To the alternation of even and odd in the number series
corresponds an alternation of Unlimited and Limit in the world. The
πάθη καὶ μέρη of the heaven are reflected in numbers (986a5), as is the
structure of the musical scale (τῶν ἁρμονιῶν τὰ πάθη, 985b32). It is not,
after all, transcendent Being that the Pythagoreans are thinking about,
but that-which-is, thought of as something coming to be, and many-
shaped: ὄντα καὶ γιγνόμενα (985b28, 990420).
Plurality and becoming: these were the problems that Parmenides
had sct for philosophy. He called both of them unthinkable and
unsayable, because they were incompatible with his basic principle,
ἔστι. The logical postulate of Truth, according to which any meaning-
ful thought or speech presupposes a Truth, seemed to require the
existence of an absolute Being, and “Truth” could only be expressed
as “Being,” ov. To the philosophers after Parmenides this seemed
105. Cornford and Raven tried by different means to show a relation with the Eleatics.
Cf. also below, ch. ΠῚ 3. The interpretation which follows is indebted to C. H. Kahn,
“The Greek Verb ‘To Be’ and the concept of Being,” Moundations of Language 2 (1966)
245-205.
08 Cf g&sbay, 32, ὁδόν ς, YOOATY, 1ORBAI7, τοψολλι,
2. The Pythagoreans according to Aristotle
irrefutable; but still, few followed him to the logical conclusion, of
denying becoming and plurality. But how was Parmenides to be over-
come? Faced by this problem, the Pythagoreans were able to enlist the
help of the technique of calculation. It must have seemed difficult even
to a Zeno to deny that the thing “‘is valid”’ or “is” which we use as
the paradigm of elementary correctness—namely, that 1 and 1 is 2,
and that 2 and 2 is 4. And yet in this procedure something apparently
new is gained out of the presuppositions, multiplicity and alteration
emerge. Out of a few fundamental figures, the numbers from 1 to Io,
there develops an inextricably complicated system. The “‘ generative”
character of mathematics broke through the inflexibility of the Eleatic
system: plurality and becoming are not unthinkable and unsayable
after all, but they can be thought and expressed in the form of numeri-
cal operations. These thus become, of course, the only legitimate form
of expression about that-which-is; and the cosmogonic myth becomes
the dry calculation of a process of division or addition. By contrast
with Parmenides or Melissus, however, this is a relaxation of the
stiffness of the denial of reality, and a new attempt to get back to
everyday reality.
One could even see the foundation of mathematical science as latent
in the principle that only mathematical expression is valid expression—
only that its content, in empirical knowledge, is scanty; mathematics
is still in its earliest stages, ringed about by all sorts of connotations
and arbitrary conventions. It was only the development of mathe-
matics from elementary arithmetic to deductive geometry, at a
later time, that made a mathematical science possible—a develop-
ment that leads, by way of Archytas, to Eudoxus (cf. below, chs. IV,
VI).
“The Pythagoreans were the first to take up mathematics . . . and,
having been brought up in it, came to believe that its principles are the
principles of existing things.”%° This is Aristotle’s psychological
explanation of the origin of the Pythagorean doctrine. Modern scholars,
too, have inclined toward psychological explanations, except that
the driving force is thought to be not habituation, but the over-
whelming impression of a discovery made: that of the numerical
proportions in musical intervals.!" On the other hand, a passage
of Iamblichus, obviously derived from Aristotle's book on the
"10 O&sba3ff; cf ch. ΝῚ τ,
11 Guthrie 1 238.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
Pythagoreans, speaks of the factual justification that the Pythagoreans
had to offer: “‘ Whoever wishes to comprehend the true nature of
actual things, should turn his attention to these things, the numbers and
proportions, because it is by them that everything is made clear.”
Number is οὐσία, that about things which can, with a claim to truth, be
expressed (Arist. Met. 987a19); nothing is known without number."*
A perfectly certain interpretation of a philosophy is impossible when
it is known to us only indirectly and mostly in the context of polemic.
Its place in the history of thought, however, can be satisfactorily
assessed when we are clear about the traditional background (the
cosmogony) and the philosophical problems current at the time
(Eleaticism).
112 Jam. Comm. math. sc. 25 p. 78.8-21. (On the deletion of the intrusive reference to
geometry, see below, ch. VI 3, nn. 2-5, with context.) Compare the passages from
Aristotle in the right-hand column.
ot δὲ Πυθαγόρειοι
διατρίψαντες ἐν τοῖς μαθήμασι
καὶ τό τε ἀκριβὲς τῶν λόγων
Σ
ἀγαπήσαντες, ὅτι μόνον εἶχεν
ἀποδείξεις ὧν μετεχειρίζοντο
ἄνθρωποι, καὶ ὁμολογούμενα
ὁρῶντες [ἔνισον] τὰ περὶ τὴν
ε f Ld + 5» ~ "
ἁρμονίαν, ὅτι δ᾽ ἀριθμῶν [καὶ
4 4 4 aw 7 ᾿
τὰ περὶ τὴν ὄψιν μαθήματα διὰ
f i ” ~
{δια γραμμάτων), ὅλως αἴτια τῶν
ὄντων ταῦτα φήθησαν εἶναι καὶ
τὰς τούτων ἀρχὰς: ὥστε τῷ βουλομένῳ
θεωρεῖν τὰ ὄντα πῶς ἔχει, εἰς ταῦτα
βλεπτέον εἶναι, τοὺς ἀριθμοὺς καὶ [τὰ
γεωμετρούμενα εἴδη τῶν ὄντων καὶ]
λόγους, διὰ τὸ δηλοῦσθαι πάντα διὰ
ὔ
τούτων. ὡς οὖν οὔτ᾽ ἐγκαιροτέρων ἂν
,
οὔτε τιμιωτέρων ἀνάψαντες
ἑκάστων τὰς δυνάμεις ἢ εἰς τὰ
πάντων αἴτια καὶ πρῶτα σχεδὸν ᾿
4 4 4 5 wm 4
ὁμοτρόπως καὶ τὰ ἄλλα τούτοις
διώριζον.
Cf. Met. 985b24: οἱ Πυθαγόρειοι τῶν
μαθημάτων ἁψάμενοι...
EN 1094b13: τὸ ἀκριβὲς... ἐν τοῖς
λόγοις
Part. an. 645a14: ἀγαπᾶν τὴν θεωρίαν...
Met. 985b31: τῶν ἁρμονιῶν ἐν ἀριθμοῖς
ὁρῶντες τὰ πάθη... 986a3: ὅσα εἶχον
ὁμολογούμενα...
985b25: τὰς τούτων ἀρχὰς τῶν ὄντων
ἀρχὰς φήθησαν εἶναι πάντων.
1023b23: λόγος ὁ δηλῶν ἕκαστον
(ὁ διορισμὸς) ἔγκαιρος. Pl. Pol. 282e.
Theophr. Met. 6b13 . . «εἰς τὰς
ἰδέας ἀνάπτων, ταύτας δ᾽ εἰς
τοὺς ἀριθμούς... 1078b22: ὧν
τοὺς Aoyous εἰς τοὺς ἀριθμοὺς
ἀνῆπτον. De an.4o4b21: τὰ δ᾽
ἄλλα ὁμοιοτρόπως.
This passage cannot have been formulated by Iamblichus himself (following Arist. Met.
985b24ff), for there is not the slightest hint of the immateriality of the numbers, which
was so important to any Platonist (cf. Comm. math. sc. p. 74.9ff). Since in general the later
tradition does not usually follow Aristotle, an intermediary source (a re-working of
οὐ “Ὁ 24, but free of Platonizing touches) would also be unlikely. Rather, since both the
preceding material in Tamblichus (below, ch. I$) and ¢hat which follows (Arist. frr.
§2~-§3, guaranteed by Procl. Iu Kuch 28.13ff and Cie. ‘Tuse. 7.69) come from Aristotle,
the answer must be that here too we have an independent fragment of Aristotle.
8 Alex. Met. 40.12-- Arist. fr. 203; cf below, ch. HD 2, and above, nm. gt.
2. The Pythagoreans according to Aristotle
Finally we come to the “table of ten opposites,’ which Aristotle
sets apart from the rest of the Pythagorean number theory which
he treats: “Other members of this same school say there are ten
principles, which they arrange in two columns of cognates”’ (986a22,
Ross tr.):
(1) limit (πέρας) : unlimited (ἀπειρον)
(2) odd (περιττόν) : even (ἄρτιον)
(3) one (ev) plurality (πλῆθος)
(4) right (δεξιόν) : left (ἀριστερόν)
(5) male (dppev) : female (θῆλυ)
(6) resting (ἠρεμοῦν) ᾿ moving (κινούμενον)
(7) straight (εὐθύ) crooked (καμπύλον)
7 ae i) | cree a
9) good (ἀγαθόν ad (κακόν
(10) square (τετράγωνον) oblong (ἑτερόμηκες)
Though the good has the second-to-last position, the arrangement
is clearly made from a normative point of view. In the Nicomachean
Ethics Aristotle speaks explicitly of a “column of goods” ; there, too,
he attributes the system to the Pythagoreans, but adds, “and Speusip-
pus, too, seems to have followed them” (1096b6). The inclusion of
movement in the same column as the unlimited or indefinite (ἄπειρον,
ἀόριστον) is alluded to by Aristotle in the general remark that “τῆς
principles in the second column, because they are negative (στερητικαῦ
are indefinite (ἀόριστοι) ᾿᾽115 This is in a context which, according to
the definite statement of Eudemus, applies especially to Plato, or
better, to the interpretation and systematizing of Plato in the Academy.
At the end of the Metaphysics, Aristotle alludes again to the “column
of the beautiful («aAdv),” sets forth his opinion of the right point of
view, and in his final sentence shows again what the real object of his
criticism is : the “‘separation”’ of the Academics.1!6
Thus the “table of opposites” is quite closely connected with
Academic doctrines; we have here a continuous transition between
Pythagorean and Platonic. There is one small suggestion of Archytas
(n. 115). Hermodorus’ report of the On the Good reveals a similar train
of thought: an initial threefold division of Being is traced back to a
1M T096b5; 1106b28f. Here the Platonic Unlimited Dyad is brought into close con-
nection with Pythagoreanism (Kriimer 347).
15 Phys, 201b25° Met. 1066a14. CF. above, αν 106; on Cael. 284b6ff, above, n. 49.
"6 jog3bitff; b27: μὴ χωριστὰ εἶναι τὰ μαθηματικά, Further, cf. 1072031, Ger. corr.
31yars, 10§4a29ff, with a reference to Περὲ ἐναντίων,
I PLATONIC AND PYTHAGOREAN NUMBER THEORY
twofold division, in which one side has “equal, abiding, harmonized,”’
(ἴσον, μένον, ἡρμοσμένον) and the other “‘unequal, moving, unharmon-
ized’ (ἄνισον, κινούμενον, ἀνάρμοστον) ..}7 The connection with Speusip-
pus is particularly close."8 Therefore it is not surprising if later
Platonists, and also pseudo-Pythagorean works, keep introducing
similar “tables of opposites.’’!!9
On the other hand, Aristotle considers it possible that Alcmaeon
already knew the “table of opposites.”” This does not mean, however,
that Aristotle dates it to the beginning of the fifth century,!”° but that he
cannot, or will not, make a positive statement on the chronology; the
table may be older or later than Alcmacon. It may be that he knew it
only from the oral tradition, passed on, later, by Speusippus. To think
in terms of polarities, of antithetical pairs of concepts, is an old human
habit.#1 The fact that there are ten pairs, however, and the inclusion of
square and rectangle, do not give the impression of being ancient; and
Alcmaeon is close to Ionian philosophers like Heraclitus. There is even
more uncertainty latent in the possibilities of interpretation than in the
chronology. Is such a rigid schema the result of true philosophical
reflection, or of a primitive way of thinking? Or is it the expression
of a strictly regimented way of life? We can see what the Platonists
made of it, but it is not a helpful foundation for a reconstruction of
Pythagorean philosophy.
117 Simpl. Phys. 247.30ff.
118 The idea that “good” does not belong at the beginning, but only appears as a later
development, comes from Speusippus (1072b3o0ff, applied to the Pythagoreans and
Speusippus in common; cf. 1075236ff, 10g1a29ff, 434: τῶν viv τινες), Cherniss, Pres. 241
n. 111. In addition, the pair one-plurality, which occupies third place in the “‘table of
opposites”’ is fundamentally important to Speusippus.
119 The following (a and Ὁ) are probably based on Aristotle: (a) Plut. De Is. et Os.
48.370e (the order, compared with Aristotle’s, 9 3 1 6 7 2 10 4 8; the ‘‘good” is at the
beginning; instead of “ plurality” he has ‘‘dyad’’). (b) Por. VP 38 (from Diogenes Anton-
ius?): monad-dyad, light-darkness, right-left, equal-unequal, abiding-moving, straight-
circular. (c) Eudorus ap. Simpl. Phys. 181.22ff: ordered-disorderly, definite-indefinite,
known-unknown, male-female, odd-even, right-left, light-darkness. (d) “‘Pythagoras”’
in Varro Ling. 5.11: finitum-infinitum, bonum-malum, vitam-mortem, diem-noctem,
status-motus. (e) Ps.-Archytas p. 19, 5-13 Thesleff: ordered-disorderly, limited-unlimited,
spcakable-ineffable, rational-irrational, binding-bondspoiling, etc. (f) “‘Eurysus,’’ Stob.
1.6.19: speakable-ineffable, ordered-disorderly, rational-irrational. (g) Philo Qu. in Exod.
2.33: odd and god-even and mortal, equality-inequality, similarity-dissimilarity, same-
different, unification-dissolution, better-worse. Cf. also Tim. Locr. 1.
120 So Raven, Pyhl tof, KR 293.1, De Vogel, Rev. philos. 1959, 34f, Guthrie I 233,
2447: on the other hand, the table is dated in the second halfof the fifth century by Zeller
460 and Philip 37.
MiG EL 1. Lloyd, Polarity and Analogy (Cambridge, 1g60), esp. 11 -φας There is a
tempting similarity to Iranian dualism, noted, as it seems, by Aristoxenus (fr. 11
Elippol, i.2.12).
3. The Later Tradition and Its Sources
3. THE LATER NON-ARISTOTELIAN TRADITION AND ITS SOURCES,
SPEUSIPPUS, XENOCRATES, AND HERACLIDES PONTICUS
The most important of the later sources for Pythagorean philosophy
are the Pythagorean Memoirs (Πυθαγορικὰ ὑπομνήματα) excerpted by
Alexander Polyhistor, the Life of Pythagoras (Πυθαγόρου Bios) excerpted
by Photius, the reports of Aétius, and (most extensive of all) those of
Sextus Empiricus.
It will become increasingly clear as we proceed that the so-called
Memoirs (or Hypomnemata) can surely not be an original Pythagorean
writing of the fourth century B.c., as Wellman and Delatte tried to
prove. After the careful study of Festugiére, we may date them with
some confidence to the end of the third century B.c.1
The life of Pythagoras which Photius read has now been attributed
by Theiler, on good grounds, to Eudorus.?
Sextus Empiricus gives four rather comprehensive accounts of
Pythagorean doctrines, of which two pairs are more closely connected
with each other than with the rest. The discussion “‘on number”’
(PH 3.151-167) is repeated in much greater detail at Math. 10.248-309.°
There is a detailed exposition of the doctrine of ideal numbers, followed
by a refutation of the concept of participation and “separation”; and
because of a remarkable agreement with Hermodorus’ account of
Plato’s doctrine of principles, the text has rightly been employed in the
reconstruction of these Platonic theories; it must derive from one
account of Plato’s lecture On the Good.4 It must be remembered,
1D.L. 8.24-33 (on the extent of the passage, see below, ch. II 4, ἢ. 4) = FGrHist
273F93 = DK 58Bia. Its authenticity was maintained by Wellmann, Hermes 1919; this
led to its inclusion in later editions of DK. Delatte (Vie 198-237) reached the same conclu-
sion independently of Wellmann. Wiersma (Mnemosyne 1942, 97ff) introduced qualifica-
tions. Zeller (III 2.103-108) dated it in the first century B.c., and was followed by Jacoby,
F'GrHist 273F93. Festugiére gave conclusive arguments against authenticity in REG 1945.
In Philologus 1961, 26, I tried, by combination with the “‘letter of Lysis,” to establish a
third-century B.c. date for the book.
2In Parusia (1965) pp. 200ff. Immisch (SBHeid 1919, 7) had sought to show that the
author was Agatharchides. Against this, see Wilamowitz, Platon II? 84.1, Ueberweg-
Pracchter 157*, and K. Reinhardt, RE XXII 763—768 (on the influence of Posidonius).
Ps.-Justin Coh. ad gent. 19 cites the book (cf. 438b33, 439419): Von Fritz (ΘΒ ΜῊ 1960, 6)
considers the word σεβαστικοί (438b19) to be of imperial date, but as early as Epicurus we
find σέβασις and σεβασμός (fr. 141. Sent. Vat. 32).
* Raven, Ἀγ Εἰ 105ff, rashly attributes all four reports to the same source. In the following
argument we need not cite more than Math. 10; PH does not add anything. At PH 3.152
the irrelevant of ἀτμοί should be emended (comparing Math. 10.252ff) to at ἄτομοι
(Mutschinann should have taken this conjecture of Pappenheim into the text).
‘Heinze, 37; Merlan, Philologus 1934, 37f; Wilpert, Hermes 1yqgi, Zwkr αι De
Vogel, Mucmosyne 1949, 209ff; Kriimer 282 Εἶν above, ch. {ας
I, PLATONIC AND PYTHAGOREAN NUMBER THEORY
however, that Sextus ascribes the whole, unambiguously, to the Pythag-
oreans, and even to Pythagoras himself.> It is scarcely possible to
determine the immediate source Sextus is following;® it is later than
Epicurus, who is cited in it, and even later than Asclepiades of Bithynia.’
A differently organized exposition, under the heading “On the
Criterion,” deals with the relationship between mathematical propor-
tion and nature (Math. 7); and the same subject is treated much more
briefly, in a different order, and with some supplementary material, in
the book “Against the Arithmeticians’’ (Math. 4).8 In the former is
found the famous citation of “Posidonius in his exposition of Plato’s
Timaeus” (Math. 7.93). The bitterly debated question, whether this
means that Posidonius wrote a commentary on the Timaeus, we may
here leave unsolved;® Reinhardt and Schmekel agree that there is much
more from Posidonius than a single sentence.!° Thus we have, in this
passage, an exposition of Pythagorcanism as it was seen by Posidonius.
It is an important question, nevertheless, whether Reinhardt was right
in singling out the explanation of the Pythagorean Tetractys (94-100)
as “4 piece of book-learning that Sextus could have gotten from
anywhere.”
This is in fact the most popular exposition of Pythagorean doctrine,
5 of περὶ Πυθαγόραν Math. 10.248, 250; cf. 255, 263, 282, 284; Πυθαγόρας 261. Similarly
PH 3.152, 157, 163. Plato’s theory of ideas is mentioned, polemically, in Math. 10.258.
It is hard to understand why Ross should say (PTI 186) that Sextus was “‘ vague as to the
authorship of this scheme.”
6 Using the difference in the derivation of line, plane, and solid mentioned in Math.
10.281f, Schmekel (Mittl. Stoa 403ff) constructed two opposed neo-Pythagorean systems,
a monistic one going back to Posidonius and a dualistic one transmitted by Antiochus.
This remains far from certain. It is true of both the “point” and the derivation of the
Indefinite Dyad from the One that they are not taken directly from Plato (cf. Wilpert,
ZwFr 174f, above ch. I 1, n. 17).
7 The citation of Epicurus at Math. 10.257 (fr. 276 Usener) comes from Sextus’ source,
since PH 3.1§2 also alludes to it. The ὄγκοι of Math. 10.252, 254 are probably those of
Asclepiades (cf. 7.201f; Theiler, Isonomia gof). Theiler, Parusia 208 suspects that Eudorus
is the source of the whole exposition; but Eudorus modifies the “ Pythagorean”’ system
by setting the One above the pair of opposites (below, n. 45).
8 7.94, cf. 4.2-3; 7.99-100, cf. 4.4-5; 7-95-98, cf. 4.6-9. The account in Math. 4 is more
“by way of brief illustration” (4.4, 10), but it alone has the statement on the number Io
(4.3; lacking 7.94 but alluded to Philo Op. 47), as well as the connection of tetractys and
soul (cf. Anat. 32).
® Cf. Reinhardt, Poseidonios 414ff; RE XXII $69, with references.
10 Schmekel, Mittl. Stoa 405ff, Reinhardt, Poseidonios 415f, 418f. The threefold division
of things into συναπτόμενα, ἡνωμένα, διεστῶτα (Sext. Emp. Math. 7.102; Reinhardt,
Kosmos 34ff) is unmistakably Posidonian; and there is a clear piece of evidence in the
mention of the Colossus of Rhodes (Sextus 7.107; Reinhardt, Poseidonios 419).
Ἡ Reinhardt, Poscidonios 416, RE XXII 725. In agreement, among others, is M.
Pohlenz, Hermes 76 (1041) 2.
3. The Later Tradition and Its Sources
which has very close parallels in Philo,!2 Theo,!* and Anatolius.4 In
Sextus the passage merely serves the purpose of clarification, not of
development of the thought; but since Posidonius comes first to mind
if we think of a possible common source of Sextus and Philo, it is
natural to ask whether Posidonius himself may not have been the one
to take over and work in a piece of “book learning.”
The elimination of the “interpolation” does make difficulties.
Reinhardt himself had trouble deciding where it begins, and in any
case this must be in the middle of a sentence." At its end lies an incon-
testably Posidonian train of thought, set off by the phrase καὶ ἄλλως
(101),1® but section 99 is introduced in just the same way, so that the
transitional flourish merely demonstrates the unity of the passage’s
structure. The controversial section is referred to twice in the sequel.
The concluding sentence (109)!” repeats a verse cited there,!8 though
one cannot simply regard it as separable from the Stoicizing com-
mentary. Further, in the Posidonian passage (rorff), there are men-
tioned as incorporeal point, line, and plane, “which we also discussed
12 Op. 47, cf. Sext. Emp. Math. 7.94; Philo 48, cf. Sext. Emp. Math. 7.95-98; Philo 49,
cf. Sext. Emp. Math. 7.99-100. The agreement in order and formulation is striking, even
though in some parts Sextus is fuller, and in some parts Philo. Philo’s mention (50) of
the game of putting a nut on top of three others to make a pyramid (καρυατίζειν, cf.
Anat. 32.3ff), provides an explanation for the word ἐπαιωρήσωμεν in Sext. Emp. Math.
7.100. (Consequently, at Math. 4.5, instead of the inappropriate ἐπιθεωρήσῃ, whose
correct use is illustrated by PH 3.154, we must read ἐπαιωρήσῃ.)---Α short summary of
the same train of thought is given by Philo V. Mos. 2.115.
18 Sext. Emp. Math. 7.95-98, cf. Theo Sm. 93.21ff. In particular the “‘ first tetractys” of
Sextus 7.95 is only comprehensible through Theo 94. 10ff. For “τς nature of the whole,”
Sext. Emp. Math. 7.93, cf. Theo Sm. 94.4; for the oath by the tetractys (Sext. Emp.
Math. 7.94), cf. Theo Sm. 94.6f; in addition the explanation τὸν μὲν παραδόντα λέγοντες
Πυθαγόραν in Sextus, and in Theo τὸν παραδόντα [Πυθαγόραν λέγουσιν. Theo’s source here
is obviously Thrasyllus (93.18 refers to the Thrasyllus quotation at 87.4; cf. Schinckel,
Mittl. Stoa 409 n. 3). Theo’s passage is also relevant in the Timaeus discussion: the ‘second
tetractys’ is the sequence of numbers in the Timaeus.
14 Pages 31f (abbreviated in Th. ar. 29.10ff); esp. p. 32 = Th. ar. 30.4-14 corresponds
almost word for word with Sext. Emp. Math. 4.6-9. Also cf. Th. ar. 29.10: πρώτη ἡ
τετρὰς ἔδειξε τὴν τοῦ στερεοῦ φύσιν, with Philo Op. 49: πρώτη yap αὕτη (ἡ τετρὰς) τὴν
τοῦ στερεοῦ φύσιν ἔδειξε.
15 Reinhardt, Poseidonios 416.3, places the end of the citation οὗ Posidonius in the middle
of section 93, but in his paraphrase (p. 418) he finds it necessary to add the concluding
sentence of 93 about number and reason. In RE XXII 725, he adds the beginning of 94
to the Posidonius citation.
16 Above, ἢ. 10.
11 κοινῷ τε λόγῳ πᾶσα τέχνη ἐστὶ σύστημα ἐκ καταλήψεων (SVF I 73), τὸ δὲ
σύστημα ἀριθμός: τοίνυν ὑγιὲς τὸ ἀριθμῷ δέ τε παντ᾽ ἐπέοικεν.
18. 7.94. In the same way, the end of section 98, and too, allude to the tetractys oath in
94. Reinhardt (Poseidonios 416 n. 1) interprets this as characteristic of the “interpolated”
passage.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
a little earlier” (104). Now since in a later passage, which is certainly
Posidonian, there are mentioned precisely the “incorporeal ideas ...
consisting in the borders of bodies,’’!® the present passage, too, will
have to be attributed to Posidonius. The continuity of the whole
passage is irreproachable: the criterion of truth for the Pythagoreans
is the ‘‘reason that is achieved from mathematics’’ (92), since like is
known from like; the next logical step is to show to what extent nature
is mathematically constructed, and thus akin to mathematical reason.?°
The solution is in the thesis that “everything is like number,’ which is
then explained in various instances. If Posidonius, 4 propos of the
Timaeus, wished to speak about the recognition of like through like,
he could not ignore the fact that in the Timaeus the soul, which recog-
nizes, is created as a number pattern, as the physical world is made up
of mathematically determined triangles. Obviously, Posidonius is in
part dependent on Aristotle, who, following a similar line of thought,
brought the Timaeus and the theory of ideal numbers into connection,
and also treated of this matter in the dialogue On Philosophy.”
Thus it remains probable that the whole section in Sextus, that
is to say the most prevalent type of exposition of the Pythagorean
theory of numbers, was transmitted by way of Posidonius.?? The
additional material in the parallel passage?’ is probably to be explained
by the fact that Posidonius was abridged twice, in Sextus’ work, in
different ways. It is important to see that not only in the attention
he pays to the Timaeus, but in details of his interpretation, Posidonius
goes back to the Old Academy.”
19 Math. 7.119 repeats almost word for word what “Posidonius in his exposition of
Plato’s Timaeus”’ had said in 93. Point, line, and plane are incorporeal in 99-100; and in
104 they as well as time can be given as examples of incorporeality.
20 Reinhardt is wrong in missing, in the “interpolation,” the relation to the problem
of knowledge (Poseidonios 416).
21 De an. 404b9ff has the same quotation from Empedocles as Sextus 7.92 (above, ch.
I 1, nn. 47-55); cf. Chalcid. 50.
22 Cf. F. E. Robbins, “‘Posidonius and the Sources of Pythagorean Arithmology,
CP τς (1920) 309-322; “The Tradition of Greek Arithmology,” CP τό (1921) 97-123;
V. DeFalco, ‘‘ Sui trattati aritmologici di Nicomaco ed Anatolio.” Riv. indo—greco-italica 6
(1922) fasc. 3~4, pp. 51-60. Robbins assumes a pre-Posidonian source; DeFalco points
out the fundamental importance of Posidonius’ exposition. These studies may be consulted
for the individual problems of filiation in the very complicated tradition; cf. also W.
Theiler, Gnomon 28 (1956) 284f, and P. Boyancé, REG 76 (1963) 82ff.
23 Above, n. &.
34 Exegesis of the Timaeus by Posidonius is also attested by Plut. De an. procr. 1023b
(a passage that would also fit into a discussion of the criterion of truth; cf. Merlan, Philo-
logus 1934, 2ttf); and Theo names Posidontas, at 103.68, 10 the course of an account of
number speculation taking off from: che ‘Vimacus. Cl Merlin, Philologus 1934, 197,
PINcoplt att, below, a. 75.
3. The Later Tradition and Its Sources
Sextus Empiricus occasionally uses the name Πυθαγόρας in addition to
Πυθαγόρειοι; but the doxographer Aétius even uses Πυθαγόρας more
often than Πυθαγόρειοι. The two coalesce into the expression of ἀπὸ
Πυθαγόρου, in the later tradition,2® but some trace of the difference
has remained: one branch never has Πυθαγόρειοι where the other
has Πυθαγόρας.
Now, we immediately notice a fact of great importance: the majority
of the reports about Πυθαγόρειοι can be confidently referred to Aris-
totle as source,*® but this is not so of even one of the reports about
Πυθαγόρας. Pythagoras is frequently named in the same breath as
Plato, Pythagoreans never.2” Thus from the external form of the
tradition itself, it is clear that alongside the Aristotelian tradition about
the Pythagorean philosophy there was another, which dared to name
Pythagoras himself and connects him closely with Platonism. And to
be sure, in discussing Sextus’ reports we constantly kept finding it
necessary to refer to the Academy.
“The first principle of all things is the Monad; from the Monad
comes the Indefinite Dyad to serve as matter for the Monad which is
cause, and from the Monad and the Indefinite Dyad come the numbers.”
So runs the beginning of the Memoirs excerpted by Alexander (D.L.
8.25). Aétius’ phrasing is similar: “Among their principles are the
Monad and the Indefinite Dyad. For him, one of the principles is
directed toward the active or formal cause ...and the other toward
2 Aét. 3.2.1: Πυθαγορείων, Stob. τῶν ἀπὸ Πυθαγόρου, Plut.; 1.15.7: of ἀπὸ Πυθαγόρου,
Plut., but Stobaeus Πυθαγόρειοι continuing from 1.15.2; see Diels’s note; 1.9.2: of ἀπὸ
Πυθαγόρου, Stob., Plut.; but Theodoret has, in more detail, Θαλῆς μὲν καὶ Πυθαγόρας...
(cf. Diels, Dox. 46); 4.14.3 belongs with 4.13.10: Πυθαγόρας, as does 1.16.1 to 1.24.3,
but 2.9.1 to 1.18.6: Πυθαγόρειοι. 1.14.2 remains problematic. (Plutarch has the correct
text, as Diels says in his note. There is a relationship to Cleanthes: Aét. 1.14.5, 2.14.2,
cf. 2.2.1; Por. ap. Euseb. Praep. evang. 3.7.4)
76 Mostly particular questions in natural philosophy: Aét. 1.15.2, cf. Arist. Sens.
439430. The material appended in 1.15.7 is simply attribution of the doctrine of Empedo-
cles (1.15.3), who was considered a Pythagorean, to Pythagoreans generally.—At 1.18.6
Aristotle is explicitly cited (Phys. 213b23, fr. 201); 2.9.1 belongs in the same context.
At 2.29.4 Aristotle is cited along with Philip of Opus (not in Rose; Ross p. 142). Cf. 3.1.2
with Mete. 345a14ff, bioff, and 3.2.1 with Mete. 342b3off; the doctrine of Hippocrates of
Chios (342b36) and his pupils is attributed to the ““Pythagoreans,” as well as the anonymous
doctrine of 34sbioff, which belongs also to Hippocrates, There remain 2.13.15 (“‘ Heraclides
[fr. 1134 Wehrli] and the Pythagoreans”’), 2.30.1 (‘some of the Pythagoreans, including
Philolaus” [DK 44A20]), and 2.22.5, which probably goes with 2.13.15. Of 10 lemmata,
6 are surely taken from Aristotle; others are from Heraclides, Empedocles, and Philolaus.
27 Pythagoras and Plato: 1.23.1, 4.4.1, 4.7.1, 4.7.5, 4.9.10, §.20.4. Pythagoras, Plato,
and Aristotle: 2.10.1, 4.20.1, 4.4.2, Pythagoras and Plato in contrast: 1.10.2-3, 4.2.3-5,
2.4.1-2.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
the passive or material.’ Further on the Monad is also called mind,
god, and good, and the Dyad is called divinity (δαίμων) and evil.?
The long passage in Sextus builds on the same foundation: “ Pythagoras
said that the first principle of existing things is the Monad... and
being added to itself. . . it produces the so-called Indefinite Dyad . . .”
(Math. 10.261; cf. 276) “. . . and the rest of the numbers were produced
from these ... The Monad had the position of the active cause, and
the Dyad that of the passive matter’’ (277). In the Photius passage the
continuity is broken, but the same general outline is discernible:
alongside a “ Monad, i.e. the one that is comprehended by intellect,”
which is distinguished from the number 1 (438b14), and which is
‘first principle of all things” (439a19), stands the Indefinite Dyad,
which is perceived “according to excess and defect” ;?° and “‘from the
Monad and the Dyad” come the numbers (439a2).
The doctrine attested in these four passages is as homogeneous in
content and terminology?! as it is flagrantly contradictory to everything
that Aristotle ascribes to the Pythagoreans. There is not a word about
Limit and Unlimited, about odd and even, about the One as a bisexual
entity, or of cosmogonic development. On the other hand, every
aspect corresponds to the system of derivation which Aristotle ascribes
8 1.3.8. This report is linked by the expression πάλιν δέ to another one: “The first
principles, he said, are the numbers and the proportions contained in them, which he
also calls harmonies, and the compounds of these two are the elements, which are called
geometrical figures (yewperpixd)”’ (cf. Cic. Acad. 2.118). Similarly in 1.10.2, “Pythagoras
posited the so-called forms... in the numbers and in their harmonies and in the so-called
geometrical figures, as inseparable from bodies.” This is confronted with Plato’s
““separation’’ (1.10.3), so that here the numbers of Pythagoras are being interpreted
as immanent. The ‘geometrical’ construction of the elements out of numerical
proportions presupposes Tim. §3b. Now the whole section begins with the φιλοσοφία
legend of Heraclides (frr. 87-88 Wehrli; Burkert, Hermes 1960, 159f€), which also had
something in it about “numbers” in the world (at least if lam. VP 59 is even partly based
on Heraclides [Burkert, ibid. 161ff]). Thus one may conjecture that this passage of
Aétius is based ultimately on material from Heraclides—who was influenced by Platonic
ways of thinking.
29 Aét. 1.7.18 with ps.-Galen Phil. hist. 35 (Dox. 618.12).
80 438b34: μονὰς μὲν yap... ἐνομίζετο ἡ ἐν τοῖς νοητοῖς οὖσα, ἕν δὲ τὸ ἐν τοῖς
ἀριθμοῖς" ὁμοίως δὲ καὶ δύο τὸ ἐν τοῖς ἀριθμητοῖς. Immisch omitted this second
sentence from text and commentary, obviously by oversight. The discrepancy between
ἀριθμοῖς and ἀριθμητοῖς cannot be removed by a simple correction. There ought to be a
three-part gradation: (1) the principles Monad and Dyad, (2) the numbers 1, 2, 3, etc.
(for this differentiation between Monad and Dyad on one hand and the one and the two
‘in the numbers [ἐν τοῖς ἀριθμοῖς}, Sext. Emp. Math. 10.276, Lydus Mens. 2.6, Philo
Qu. in Gen. 4.110), and (3) countable objects, ἀριθμητά. (Detailed differentiation of num-
bers and countables in Sext. Emp. PH 3.157ff, Math. 10.2846, 4.1 Theo Sm. ty.1sff.)
31 To be sure, the Indefinite Dyad is derived from the Monad at Sext. Emp. Math.
10,261, and less clearly at Hypo. 25, though there is no suggestion of this in Photius
and Aétius, any more than in the reports about the On the Good, CE above, mn. 6.
3. The Later Tradition and its Sources
to Plato and his pupils, in distinction from the Pythagoreans. The pair
of Monad and Indefinite Dyad is Platonic,®* the concept of the In-
definite Dyad is purely Platonic, the idea of excess and defect (great-
and-small)83 is explicitly characterized by Aristotle as peculiar to Plato,*4
the matching of form and matter is Platonic-Aristotelian, like that of
active and passive,®®= and Aristotle labored in vain to apply these
categories of his to the “so-called Pythagoreans.’’*
The one question remaining in abeyance is that of “separation”’
(ywpiouds)—hardly surprising, since we are dealing with Hellenistic
and largely Stoicizing tradition. Once Aétius expressly denies separation
in the Pythagoreans,®’ but in another paragraph he speaks of “ partici-
pation” (μετοχή),38 and in Sextus too the concept of participation plays
a central role in both exposition and criticism.®® In any case, all agree
that the causes are, according to Pythagorean teaching, incorporeal*°—
a formulation that Posidonius, too, could accept. There is thus
unanimity that the sensible world is derived from higher principles,
82 In the reports on Plato the word is ἕν, though Xenocrates fr. 15 has Movas.
33 Anon. Phot. 438b35ff, Sext. Emp. Math. 10.263ff, esp. 275. In Aétius and in the
Hypomnemata the Indefinite Dyad is not closely described.
34 Met. 987b26f.
3 Por the concept of ὕλη Pythagoras is grouped with the Stoics at Aét. 1.9.2: “Thales,
Pythagoras, Anaxagoras, Heraclitus, and the whole flock of Stoics (ὁρμαθός, Theodoret)
say that matter can be turned and altered and transferred, and that all of it is always
flowing and intermingling.” Cf. Hypomn. 25, on the elements; Aét. 1.24.3: ‘‘matter is
passive”; cf. 1.16.1; Sext. Emp. Math. 10.277, 9.366, etc. This too suggests the Timaeus;
cf. Arist. fr. 207 (below, n. 164), Xenocrates fr. 28 (below, n. 124). At Aét. 1.23.1 the
definition of movement as “a difference or otherness in matter” is ascribed to Pythagoras
and Plato; Eudemus fr. 60 attributes this definition to Plato, and distinguishes him in this
from Archytas (above, ch. J 2, ἢ. 106).
36 Above, ch. I 2, n. 95.
7 1.10.2; cf. above, n. 28. In the anonymous Prolegomena in Platonem 5.36-38 (p. 201
Ilermann) it is stated that Plato put his ideas ἐν τῷ παραδειγματικῷ, Pythagoras and
Aristotle ἐν τῷ ποιητικῷ.
8.7. 11,3.
49 Sext. Emp. Math. 10.261: “(τς monad) by participation in which each thing is
called one (ἕν λέγεται). From a similar source, Theo 21.2 has “(the monad) . . . for by
participation in it each thing is called one.” (In the sequel this is contaminated with
different matter. The whole may come from Moderatus, to whom Stob. 1 proem. 8
attributes the beginning, and Theo the passage 18.3ff.) Its model is to be found in Eucl.
ἡ def. τὶ “monad: that according to which each existing thing is called one.’ The “ Pythag-
orean” sources introduce the idea of participation, and then Sextus’ criticism (Math.
10.284ff) concentrates on this point.
40 “The first causes are incorporeal, “‘Aét. 1.11.3; more fully Sext. Emp. Math. 10.250ff,
“οἵ, 104; “the monad... that is among mentally apprehended things,” Anon. Phot.
438b34; “all bodies come from the monad,” 439a23f. In Hypommn. 25, the “perceptible
bodies’? only come into being after point, line, plane, and (geometrical) body. Philop.
(De an. 82.19) significantly names Pythagoreans and Xenocrates (fr. 29) together as
proponents of “incorporeal principles” (ἀσώματοι ἀρχαί). :
41 Sext. Emp. Math. 7.119; cf above, a. τὸ. .
I, PLATONIC AND PYTHAGOREAN NUMBER THEORY
and there is presupposed an ontological gradation like that of Plato’s
system of derivation. This too contradicts the statements of Aristotle,
according to which the “‘so-called Pythagoreans”’ did not go beyond
the sensible.*2 What the Hellenistic tradition presents as the philosophy
of Pythagoras is, according to Aristotle's positive pronouncement,
Platonism and not Pythagoreanism.
This tradition about Pythagoras, inconsistent with Aristotle,
apparently came to dominate the field completely.** A few indications
will show this well enough. The Platonist Eudorus of Alexandria
(first century B.c.) says that, “‘according to the highest logos,” the first
principle of the Pythagoreans was the One, and “according to the
second logos,’ an “opposite nature’ along with the One. The opposi-
tion is developed with a “table of opposites,” and summarized in the
formula “One (or Monad) and Indefinite Dyad.”’*
Moderatus transposes the Pythagorean number doctrine into
logical-conceptual language,** but starts from the presupposition that
One and Dyad are the principles. He elicits from the Parmenides a
doctrine of the stages of the One which Plato is supposed to have taken
from the Pythagoreans.*? According to Hippolytus, Pythagoras
posited the “unbegotten (ἀγέννητος) Monad”’ as first principle, from
which the Dyad and the other numbers had their origin.4® On the
other hand Numenius ascribes a radical dualism to Pythagoras: the
42 Above, ch. 12, ἢ. 15. |
48 Theo says, 20.5ff, that “the later ones” had introduced Monad and Dyad as principles,
but that “τῆς disciples of Pythagoras” (ot ἀπὸ Πυθαγόρου) had posited ‘‘all the terms
set out in series, by virtue of which even and odd are conceived,” i.e. the system of number
concepts. The sharp distinction between τὰ ἐν αἰσθητοῖς τρία and τριάς, between ἀριθμητόν
and ἀριθμός (above, ἢ. 30), shows that this is not in agreement with Aristotle. Rather,
these are Moderatus’ own views (cf. Por. VP 49-51). The historical form of his statement
is obviously based on the accidental collocation in the doxography (Aét. 1.3.8; above,
n. 28).—Limit and Unlimited became important again especially after Porphyry’s com-
mentary on the Philebus; cf. lam. Comm. math. sc. pp. 12ff, and passim, Procl. In Encl.
5.14ff and passim; Nicomachus (Ar. 2.18.4) brings in Philolaus (below, ch. III 1).
44 Collected by Zeller I 465ff; cf. De Vogel, Pythagoras, 207ff. See esp. Plut. V. Hom.
145 (on the source question, Diels, Dox. 88ff), Nicom. Th. ar. 11.16, 12.9ff (the Indefinite
Dyad as ἔλλειψις καὶ πλεονασμός and as ὕλη), Por. VP 38, Iam. In Nic. 61.21. Cf. also
Festugiére, Rév. IV 18ff.
45 Apud Simpl. Phys. 181.1o0ff. (Theiler, Parusia pp. 205-208; H. Dorrie, “‘ Der Platoniker
Eudoros von Alexandreia,” Hermes 79 [1944] 25-39). The distinction of a highest One
above the pair of opposites comes from PI. Parm. 137c-141d; cf. Dodds, CQ 1928,
135 Eudorus also looks back to Xenocrates and Crantor (Plut. De an. Procr. 1013b).
46 Por. VP 48-51; cf. above, n. 43.
Por. ap. Simpl. Phys. 230.34ff.
4" Ref) 1.2.2, 1.2.6, 4.51.4, 6.23.0: “the principle of all things is unbegotten ... the
Monad, but the Dyad was begotten.” Here the difference between the Indefinite Dyad
and the number 2 is obliterated clearly a Gnostic interpretation,
3. The Later Tradition and Its Sources
Indefinite Dyad (duitas indeterminata) was coeval with the divinity, the
Monad, and the relation of the two is that of form and matter (species
and silva).4® It is clear that even the more detailed expositions of the
neo-Pythagoreans rely on the same tradition; the unmistakable
Indefinite Dyad shows that the whole construct 15, if Aristotle was
right, ἴδιον Πλάτωνος.
In fact, people did realize the inconsistency with the reports of
Aristotle. This appears not only from the polemics of the neo-Plato-
nists,°° but from a curious document, attributed to Theano, who was
usually known as the wife of Pythagoras ;™
I have learned that many of the Greeks suppose Pythagoras said
that everything came to be from number. This statement, however,
involves a difficulty—how something that does not even exist is
even thought to beget things. But he did not say that things came to
be from number, but according to number. For in number is the
primary ordering, by virtue of whose presence, in the realm of
things that can be counted, too, something takes its place as first,
something as second, and the rest follow in order.”
Here we have the neat separation of ἀριθμός and ἀριθμητόν, the
concept of μετουσία (‘‘presence” above), in almost verbal agreement
with formulations in Sextus and Theo,* and in addition direct polemic
ayainst a tradition according to which things come to be “from
numbers.” “ Many of the Greeks”’ have learned this false representation
of the Pythagorean doctrines; here there is unmistakable polemic
against Aristotle.64 The method is subtly indirect: an “original”
document is witness against his interpretation. And who is qualified
to offer authentic exegesis, if not Pythagoras’ wife and student herself?
ΤῸ be sure, Syrianus was in a still more favorable situation; he read
the ‘Iepés λόγος of Pythagoras himself, which named Πρατεύς and Avds
Chalcid. 293ff= fr. 14ff Thedinga; cf. J. C. M. Van Winden, Calcidius on Matter
(l.ciden, 19§9) 103-121.
δ Above, ch. I 2, nn. 22-23.
δ Cf. below, ch. II 2, n. 28.
δὲ Stob. 1.10.13 = p. 195.12 Thesleff. It is a remarkable idea, that number “‘does not
even exist,” in blatant contradiction of all neo-Pythagoreanism and Neoplatonism. The
unhesitating equation of being and corporeal being shows that the passage is Hellenistic
in date.
δ Above, ἢ. 39.
Probably against the lost writings on the Pythagoreans rather than the didactic
works, which can scarcely have had much influence outside the school. The word “beget”
wa barbed reference to Aristotle’s accounts (Met, τόρται 26),
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
as first principles and made number, the “ruler of shapes and forms,”
the basis of the “‘origin of all things.’”’*> Aristotle, accordingly, must
have been mistaken... This is how the gap between the different
traditions created the demand for apocrypha.
It is possible that the Pythagoras items of the Vetusta placita were
“reconstructed” in the circle of Posidonius or Antiochus, with the
help of apocryphal writings.** But these are only intermediate members
in a chain of tradition that goes back further, to weighty authorities.
Theophrastus remarks in his Metaphysics, on the problem of the good:
Plato and the Pythagoreans make the distance (ἀπόστασιν) a great
one, but they think everything wants to imitate it. And yet they set
up a kind of antithesis, of the Indefinite Dyad and the One. In the
former resides the Unlimited, the disorderly, and, so to speak, all
kinds of formlessness, yet the nature of the whole would not be
possible without 1.57
Theophrastus’ starting point is that Plato and the Pythagoreans,
both, posited a “good” at a “great distance’’ from nature as a whole.
Everything strives toward it, “and yet” they cannot get along without
an opposite principle. The opposition is explained as that between the
One and the Indefinite Dyad, in which all that is unbounded and
unordered, by its nature, is pent up. Both the term Indefinite Dyad, the
whole idea of a principle “in itself,” and the opposition of “formless-
ness” and the form-giving One are according to Aristotle Platonic
and not Pythagorean; still Theophrastus sees the two as a unity.*®
So the non-Aristotelian tradition, which is orthodoxy for later genera-
55 Syrian. Met. 9.37ff, 123.1ff. On the reconstruction of this Sacred Discourse, see Delatte,
Litt. 191ff; p. 164.21ff Thesleff.
56 Diels (Dox. 181), following Zeller (I 467ff), derives Aétius’ testimonies, so far as they
are not from Aristotle, from the pseudepigrapha. (On Posidonius, cf. below, n. 133).
57 Met. 11a27ft. The Ross—Fobes translation takes ἀπόστασιν as “the distance between
the real and the things of nature,” but the point of the whole thing is the discussion of
ἀγαθόν and κακόν. Speusippus has it that ἀγαθόν is σπάνιον and the κακόν is greatly
preponderant (11a18ff); Plato and the Pythagoreans make the distance between the two
great, though the second principle is indispensable in the world, ἐσομοιρεῖν ἢ καὶ ὑπερέχειν,
1106. Theophrastus sets his own conviction against both of these, that ra μὲν οὖν ὄντα
καλῶς ἔτυχεν ὄντα (11a25f).—Eudemus, too, after first distinguishing the two, compre-
hends both Plato and the Pythagoreans under the concept of the indefinite (above, ch. I
2, n. 106).
58 Theiler, Parusia p. 205, reads καίτοι «τοῖς» καθάπερ ἀντίθεσίν τινα ποιοῦσιν, and
opines that this formulation does not include the Pythagoreans. However, it also does not
exclude them, and in content the καίτοι sentence belongs very closely with the preceding.
The ἀγαθόν is the model of everything real, ‘and yet"’ (read ὅμως tbs instead of ὅλως)
it is not sufficient by itself; the opposite principle is equally necessary. (CE below, n. 66.)
3. The Later Tradition and Its Sources
tions, is already present in Theophrastus, even though he speaks of
Pythagoreans and not of Pythagoras. His contradiction of Aristotle
may not be explained away as concision of language or hasty com-
position.°?
Since Theophrastus was surely only a transmitter of this conception
of Pythagorean doctrine, he must be dependent on predecessors who
went still further than Aristotle®* in connecting Plato with Pythagorean-
ism—to the point of identification. The only candidates would be
Plato’s immediate pupils, the members of the Old Academy. As a
conjecture, this suggestion was made many years ago;* the proof came
to light with a fragment of Speusippus which was first published in
1953. It had made its way into Proclus’ commentary on the Parmenides,
whose final portion is only preserved in the Latin translation of William
of Moerbeke.
For the good neo-Platonic thesis that the One is still higher than
Being, Proclus refers to what Speusippus had reported as the opinion
of “the ancients” (“tamquam placentia antiquis,” 1.6. ὡς ἀρέοκοντα
Tots παλαιοῖς) :
le unum enim melius ente putantes et a quo le ens, et ab ea quae
secundum principium habitudine (i.e. καὶ τῆς κατ᾽ ἀρχὴν ἕξεως)
ipsum liberaverunt. existimantes autem quod, si quis le unum ipsum
scorsum et solum meditatum sine aliis secundum se ipsum ponat,
nullum alterum elementum ipsi apponens, nihil utique fiet aliorum,
interminabilem dualitatem entium principium induxerunt (1.6. τὴν
2. , ~ νΝ > ‘ 2 62
ἀόριστον δυάδα τῶν ὄντων ἀρχὴν εἰσήγαγον).
It is clear that as far as Proclus could tell, Speusippus was speaking
simply of “the ancients (of waAaoé).’’6 But there is only one possible
interpretation of this. Speusippus is not presenting his own system, in
59 This is Zeller’s proposal. Speaking of Theophrastus, he says (I 472 n. 4), “... in the
whole report Pythagorean and Platonic material is combined in such a way that it appears
impossible to determine, from it alone... what belonged to each.” Aristotle does say,
however, that the Indefinite Dyad is not at all to be attributed to the Pythagoreans.
"0 Met. 987a30ff (above, ch. I 2).
“1. ΟἿΣ Frank 260.1, though he overlooks the difference from Aristotle.
82 Plato Latinus II: Parmenides: Procli commentarium in Parmenidem interpr. G. de Moerbeka,
cd. Ro. Klibansky and C. Labowsky (London, 1953) 38ff- Klibansky reconstructs the
Greek text, p. 86. He suspects that the source is Speusippus’ On Pythagorean Numbers
(tr. 4 Lang), with Nicomachus as intermediary; but this cannot be proven. Merlan has
shown (PiNeop! 96-140) that lamblichus, in Comun. math. se. 15.6ff, was able to use
Speusippus in detail, perhaps through some intermediary.
48 Proclus repeats, at the end of the quotation: “testatur et iste hance esse antiquorum
opinionem.”
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
which the Indefinite Dyad was replaced by the concept of “plurality
(πλῆθος),᾿᾿ along with other modifications. But he cannot be using
the expression “the ancients” of Plato cither, even disregarding the
plural form; he only outlived Plato cight years. Therefore Speusippus
was referring to the Pythagorcans, and possibly thinking of Pythagoras
himself. Plato himself speaks of Pythagorean views as an “ancient”
revelation.®
Speusippus then, quite contrary to the statements of Aristotle,
attributes to the “ancient” Pythagorcans the opposition of “One” and
“Twoness,” the concept of the Indefinite Dyad, and in general a
principle αὐτὸ καθ᾽ αὑτὸ καὶ χωρὶς νοούμενον (“ipsum seorsum et solum
meditatium’’). The line of thought is so close to that of Theophrastus,
who was a generation younger,® that we can no longer harbour any
doubt as to the source from which the latter acquired his non-Aristote-
lian conception of Pythagorean doctrine. Plato’s nephew and successor
claimed that the basic thought of the Platonic doctrine of ideal numbers
was Pythagorean.
Speusippus was not alone in this trend. Xenocrates, too, wrote
ITv@ayépeca,®’ and we have one statement about Pythagoras from his
pen: “Pythagoras, as Xenocrates says, discovered that the musical
intervals did not originate without number, either.’ Xenocrates, then,
attributed to Pythagoras himself a specific scientific discovery about
numbers in music, but even more than this: the word καί (“ either,”
above) indicates that he traced to him some other number theory as
well.6® Now Xenocrates interpreted the origin of the world soul in the
Timacus as derivation of number from One and Indefinite Dyad,7°
and the connection of number and music, which according to Xeno-
crates Pythagoras also discovered, is definitely presupposed in this
64 Speusippus did, however, come out for placing the One above Being (fr. 34e = Arist.
Met., 1092a14ff, and the testimony discovered by Merlan, PlNeop! 96ff, in lamblichus).
A point of departure for this line of thought can be seen in PI. Rep. sogb, and better in
Plato’s Parmenides; cf. Dodds, CQ 1928, 129f.
85 Phib, 16c; below, ch. 14.
8 The elevation of the One in Speusippus corresponds to the “great distance” of
Theophrastus. Both emphasize the necessity of the second principle, without which
“nihil utique fiet aliorum,”
Met. 1088b3 5FF.
67 1).L. 4.13.
88 Πυθαγόρας, ὥς φησι Ξενοκράτης, εὕρισκε καὶ τὰ ἐν μουσικῇ διαστήματα οὐ χωρὶς
ἀριθμοῦ τὴν γένεσιν ἔχοντα, tr. y Heinze Por. In Prof. χο ΕΣ It is unlikely, in spite
of Heinze sft, that more than this sentence goes back to Xenocrates; cf below, ch. Vor.
ΟΝ Sext. Emp. Math. 10.260, who gives it as Pythagorean teaching that ἡ ἁπλῆ
γραμμὴ οὐ χωρὶς ἀριθμοῦ vevonrar. ..
Pr OB EE | Plat. Dean. proer, torad-c; following this, Tim. Locr. gse.
+ ? ~ . ᾿
οὐχ οἷόν τε ἄνευ ταύτης τὴν τῶν ὅλων φύσιν ..., as ἴῃ Arist.
3. The Later Tradition and Its Sources
section of the Timaeus.”! Furthermore, Xenocrates developed from
the Timaeus his definition of soul as “number moving itself,”’”? and
precisely this definition of soul is ascribed by the doxographers to
Pythagoras,” as well as the doctrine of the One and the Indefinite
Dyad. The later tradition about Pythagoras is largely based on the
exegesis of the Timaeus by Xenocrates, who understood the ideas
contained in Plato’s dialogue as the teaching of Pythagoras.
Speusippus, too, interpreted the Timaeus and developed a definition
of soul out of it,”4 which was taken over by Posidonius,’> namely that
the soul is ἰδέα τοῦ πάντῃ διαστατοῦ, and Posidonius adds κατ᾽ ἀριθμὸν
συνεστῶσα ἁρμονίαν περιέχοντα. Posidonius interprets the Timaeus as
evidence for Pythagorean doctrine,” and in this too he is following
Speusippus, for the new fragment is proof that the latter saw Platonism
and Pythagoreanism as a unity.
A third writer deserves mention in this context, Heraclides Ponticus.
He attributed to Pythagoras the invention and definition of the word
φιλοσοφία, and this ascription made its way, via the doxographers,
into all the ancient handbooks. Yet it is disproved by the semantic
history of the word. It had meant close acquaintance and familiarity
with σοφία; and Plato was first to define it as in insatiable striving,
and set it in opposition to the possession of wisdom. This was after
the Sophists and their claims had roused popular animosity.” Heraclides
probably combined with this a number theory that took its direction
from the Timaeus; at least he ascribed to Pythagoras the sentence, “Τῆς
knowledge of the perfection of the numbers of the soul is happiness.”’”*
Interpretation of the Timaeus and the orally transmitted theory
τι Cf. below, ch. V 1.
ΤΣ Plut. De an. procr. 1012d-e, and Xenocrates frr. 60-65.
73 Aétius 4.2.3, from which is derived Theodoret 5.17. Also Nemesius 102 M., Meletius
An. Ox. ΗΙ 146.30; cf. Cic. Tusc. 1.20 Aétius adds that Xenocrates is following Pythagoras
(4.2.4, from Stob.).
74 Fr, 40 Lang = Iam. ap. Stob. 1.49.32; cf. the discussion of Merlan, Philologus 1934,
197; Cherniss, Plato so9ff; Merlan, PlNeop! 36ff. Cherniss sets out the relationship to
the Pimacus.
τὸ Plut. De an. procr. 22.1023b, on which see Macrob. Som. Sc. 1.14.19, and the “ Platon-
ic’ definition of soul at 1D.L. 3.67. On the addition made by Posidonius, cf. Sext. Emp.
Math. 4.6-9 and Anat. 32. Th. ar. 30.4ff (above, n. 8, 14, 24; Merlan, Philologus 1934,
107}.
τὸ Reinhardt has to concede this, since the citation of Posidonius at Sext. Emp. Math.
7.93 is made in the section on the Pythagoreans.
7 Burkert, Hermes 1960, τ ἢ, cf. M. Landfester, Das griechische Nomen φίλος und
swine Ableitungcn (Hildesheim, 1966).
™ Heraclides fr. 44 Wehrli (ἴσην, Al. Strom. 2,130.3; cf. above, n. 28,
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
of ideal numbers combine, for Plato’s pupils (including Aristotle),”®
into a system which for the Platonists is regarded as the doctrine of
Pythagoras, which they themselves follow, with more or less radical
modifications; Aristotle, on the other hand, characterizes the same
system as Platonic and contrasts it with other, obviously older, less
advanced pronouncements which he calls the doctrine of the “ Pythago-
reans.” The later tradition, beginning with Theophrastus, follows the
Platonists, and to a great extent forgets the reports of Aristotle. Not
completely, to be sure: that which Aristotle ascribes to the “Pythag-
oreans” in general, appears in the doxographical reports under the
name of Philolaus.
This relationship of the different branches of tradition can also
be shown in the further development of the doctrines. The derivation
of the world from number takes place, according to ‘“‘Pythagoras,”
by the development of the point (“a monad having position”’) out of
unity,®° and then of the perceptible world from the point in the hierar-
chical succession line, plane, solid.8! As a rule the number 2 is assigned
to the line, 3 to the plane, and 4 to the solid.8? Sextus emphasizes the
difference from another derivation, according to which the line comes
to be through a continuous movement (ῥύσις) of the point, the plane
through movement of the line, and the solid through movement of
the plane. Here there is no need for any other “first principle’’ than
the One.®? Yet the two ideas get mixed together.*4
We have shown that this derivation of the physical world is firmly
anchored. in the Platonic number theory;®* it remains to show that
none of all this is ascribed to Pythagoreans by Aristotle, and that it
even contains contradictions of what is—according to Aristotle—
Pythagorean.
79 Cf. connection of the Timaeus with the ἄγραφα δόγματα at Phys, 209b11ff, and the
sequence Timaeus—ideal numbers, De an. 404b18ff, which is found again in Posidonius
(above, n. 21).
8 Procl. In Eucl. 9§.21ff, Schol. Arist. Met. 401a4 (as Pythagorean), and the passages
cited in n. 82.
"1 Fypomn. 25, Anon. Phot. 439aroff. (Skipped by Aét. 1.3.8.)
#2 Sext. Emp. Math. 10.278ff, 7.100 (cf. Philo Op. 49: Posidonius). Further, inter alia
Philo Op.98, Κ΄. Mos, 2.115, Theo Sim. 97.17, 100.20, 101.11 (with variations 96.9ff), Nicom.
Th. ar. 20.9f, 23.01ff, Procl In uel. 97.1.7ff, 114.25f; below, ἢ. 120; ch. ΠῚ 2, ἢ. 41.
83 Sexe. Emp. Math. to.28if, cf 7.99 (Posidonius); 3.10f; 9.430; Eratosthenes ap.
Sext. Emp. Math. 3.28, cf. Theo Sm. 83.21; Simon Magus ap. Hippol. Ref. 4.51.3, «ἢ
¢.9.5; Procl fa ‘Fin ΠῚ 1.37.2. On Schinekel’s views, above, ἢ, 6.
4 Thas in Posidonius, already, che “nut game” and the “flowing” obviously belong
together, Cf Sext. Emp. Math. 7.996, Philo Op. qgf, Decal. 25, Plut. 1} 1: yoced,
® Above, ch. ΠῚ, aun. 41.4.4.
3. The Later Tradition and Its Sources
In his philosophical glossary Aristotle takes over as valid the definition
of the point as “a monad having position’’®* and mentions it often, from
the point of view of * separation,” as a formula used by the Platonists. 8”
He never connccts it with Pythagoreans; and he cannot do so con-
sistently with his allegation that the “units’’ of the Pythagoreans
possess ‘“‘magnitude,”’8* for “point” and “magnitude” are mutually
exclusive terms. For this very reason the derivation of the physical
world through the series surface, line, point cannot belong to these
Pythagoreans; the world of perceptible bodies is for them the only
reality, beyond which they do not go.8* Thus Aristotle never connects
the line-plane-solid formula with Pythagoreanism, but always with
Platonism.®° Indeed there is one passage in which he clearly points
out the difference. In his De caelo he explicitly disagrees with all those
who “make up all bodies from planes’’ (299a1ff), and an express
reference (300a1) shows that he has the Timaeus especially in mind.
He asserts that to be consistent one must carry such an analysis further,
to the point, not only as far as the ἄτομος γραμμή (299a5ff), and this is
directed against Plato and Xenocrates.®! The refutation revolves about
the problem of weight, which remains unexplained in these geo-
metrical constructions; and he appends the casual remark, “and the
same thing happens to those who construct the world out of numbers:
for there are some who see nature as composed of numbers, like
some of the Pythagoreans”—here too the problem of weight is
untouched.® The Pythagoreans who “compose the world out of
numbers” are thus contrasted with ‘those who compose bodies of
8 Met. 1016b24ff; cf. De an. 409a6, An. post. 87a31ff, 88a33, Met. 1084b25.
87 Phys. 227427: “1, as they say, the point and the monad have a separate existence of
their own,” Met. 1069a12, 1077b30, 1084b23 (“1 number is separable.” bz). Top. 108b7ff
applies to Speusippus (cf. below, n. 105); against attribution to the Pythagoreans, Heidel,
AJP 1940, 27, n. “6, Wilpert ZwFr 218 n. 46.
88 Above, ch. J 2, n. 18.
8 Above, ch. I 2, ἢ. 15.
90 The evidence for Speusippus and Xenocrates, above, ch. I 1, nn. 38-39. That the
πέρατα doctrine of Met. 1028b16ff is not applied to the Pythagoreans was shown above,
ch. 1 2, nn. 77-81. The passage Met. 1036b12ff is almost always thought to refer to the
Pythagoreans: καὶ ἀνάγουσι πάντα εἰς τοὺς ἀριθμούς, καὶ γραμμῆς τὸν λόγον τὸν τῶν δύο
εἶναί φασιν. (Ps.-Alex. Met. §12.23ff—‘‘two’’ as τὸ πρῶτον διαστατόν, line 37.—Ross, Met.
Il 202 sees the source as Aristotle’s lost book. KR accept it [as pre-Parmenidean!], no. 317
[ἡ 316]. Also called Pythagorean by Zeller I 511 n. 1; von Fritz, ABG 1955, 83; Cherniss,
Pres. 22§; Guthrie I 257.) One reason is that this is set up against of τὰς ἰδέας λέγοντες;
but it is Speusippus who does not believe in ideas, but only in mathematical numbers. In
the context, the point is the separation of form and matter (χωρίζειν, 234, b7; εἶδος, b2),
so that the Pythagoreans cannot be meant.
" Above, ch. Pt, τ. 17.
Cael. gooal4ff. CA above ch. 12, 1. 19.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
planes” (299a3); they are united only in a common omission, their
failure to face the problem of weight.”
The related thought, that the point in movement makes the line, the
line in movement the plane, and the plane in movement the solid,™
appears in Aristotle’s polemic against Xenocrates. He seeks to reduce
the latter’s definition of soul to absurdity by pointing out that accord-
ing to these geometrical doctrines—introduced with the words ἐπεί
φασι (409a4)—even the “self-moving number” would have to consist
of lines.9° In order to demonstrate a contradiction, Aristotle seizes upon
a statement of the opposite party—the Platonists.
Archytas had evidently gone deeply into stereometric problems;
Aristotle’s statement that “the Pythagorcans”’ called a surface χροιάϑδ
(Plato is said to have introduced the term ἐπίπεδον) may belong
in this context. According to Diogenes Laertius, Archytas introduced
the idea of movement into geometry,*® and in fact Archytas’ solution
of the “Delian problem,”’®® reported by Eudemus, makes good use of
such concepts: a right-angled triangle, revolved about one leg, produces
a cone, the end of a straight line, when turned about, describes an arc,
etc. Archytas, according to Eudemus’ evidence, also concerned himself
with the explanation of movement in general.?°° Thus the conjecture
secms justified that this comprehensive treatment of geometrical
magnitudes by the use of the idea of movement comes precisely from
83 Bywater (31) already perceived this difference, and used it against the authenticity of
Philolaus fr. 12.
94 Cf. above, n. 83.
96 De an. 40gaiff; cf. Cherniss, Pres. 389, Plato 396 ἢ. 322, Theiler, Arist. 101, Guthrie
Ι 262ff. Whether Aristotle is playing off the conception of Speusippus against Xenocrates
(Cherniss), or whether Xenocrates could have spoken in this way himself (Theiler refers
to fr. 39, though it is “not quite uncontaminated”’)—this is a question we need not answer.
90 Sens. 439230, whence Aét. 1.15.2; cf. Arist. Met. 1og1a16 and Nicom. Th. ar.
25.15: ταὐτὸν ἐν μονάδι, ἕτερον ἐν δυάδι, χροιὰ ἐν τριάδι, σῶμα ἐν τετράδι. The latter,
however, is obviously tinged with archaism, and not genuine tradition; it takes its
origin from the doxographical report. The opposition of ταὐτόν-ἕτερον originates in the
Platonic dialectic.—Further, cf. Hero Alex. Deff. p. 20f Heiberg, Aristid. Quint. p. 110.
25 Winnington-Ingram.
97 1).L. 3.24. The concept of ἐπίπεδον is presupposed by Plato, Meno 76a and Pol. 299e;
in a different sense Tht. 173¢ — Pi. ἔτ. 202, as well as Democr. fr. 11d Αἰτίαι ἐπίπεδοι (“ter-
restrial,” as distinguished from t1c “aerial”; B1§5 is not a direct quotation). The idea
of χροιά is alluded to at Pl. Meno 75b-c. C. Mugler, ‘Sur l'histoire de quelques définitions
de la géometrie grecque, II: La surtace,” AC 27 (1958) 76-91, presents an ingenious but
N-founded reconstruction; he puts the ypoid idea earlier than Parmenides.
% 83 (referring to the problem of doubling the cube): πρῶτος τὰ μηχανικὰ ταῖς
μαθηματικαῖς πρυσχρησάμενος ἀρχαῖς μεθώδευσε καὶ πρῶτος κίνησιν ὀργανικὴν
διαγράμματι γεωμετρικῷ προσήγαγε.
"»" βάσις ἢν τ4ι Webrli DK 474ι.1.4.
1 Above, ch. 12, ἢν 106.
3. The Later Tradition and Its Sources
Archytas.° Here, too, Archytas went further than the older Pythag-
oreans° It is true that geometric figures had also been represented
with pebbles: 3 is an ‘‘odd line’’—or a triangle, or a “gnomon,”
9 is an “odd square” ;!° but here the strict derivation of spatial shapes
(γεννᾶν) was not yet felt as a problem. What ontological status Archytas
gave to the series point, line, plane, solid, can hardly be determined .1%4
Aristotle, in any case, attributed the derivation series point, line,
plane, solid, to Platonism, not to Pythagoreanism, and in this differs
from the later tradition, which saw in it the teaching of the Pythag-
oreans, and indeed of Pythagoras himself. Once more we suspect
the pupils of Plato as the source of the non-Aristotelian tradition,
and especially Speusippus, since the “point” always plays a role in these
“Pythagorean” accounts, and once again we find the direct proof:
in his book On Pythagorean Numbers, from which Nicomachus has
preserved us a considerable excerpt, Speusippus ascribed to them this
very concatenation: ‘‘For 1 is the point, 2 the line, 3 the triangle, and
4 the pyramid.” Eva Sachs already had guessed that this book was
one of the most important sources of the later tradition about Pythag-
oras.!°7 Whether or not the newly discovered fragment of Speusippus
about the Pythagorean “principles” came from the same book, in
any case it is with Speusippus that the tradition begins which contradicts
the reports of Aristotle.
It is now no longer surprising to find that the series of types of
101 Frank 370f, accepted by Wilpert ZwFr 174.6. Frank goes on, in an imaginative
vein, to construct a “dynamic system of Archytas” (124ff), but far exceeds the limits of
what can be known of the matter. Cf. Cherniss, Pres. 388f.
102 Cf, above, ch. I 2, nn. 104-106.
103 Cf. Philolaus A26.
104 “Pure” mathematics, divorced from ontological problems, is hardly to be assumed
in Archytas. Arnobius 2.9 (Dox. 172) names Pythagoras and Archytas: ““causam in
numeris ponit.” But Plato’s philosophy, through dialectics and theory of ideas, was
bound to change whatever he took over. The connection of the numbers 2, 3, and 4 with
line, plane, and solid, which seems more primitive than the motion idea of Archytas,
might be an attempt to derive these concepts by purely logical means, without introducing
motion.
15 For Speusippus the point is οἷον τὸ ἕν (Arist. Met. 1085333); ὅτι στιγμὴ ἐν γραμμῇ
καὶ μονὰς ἐν ἀριθμῷ" ἑκάτερον γὰρ ἀρχή (Top. 108b26; cf. above, ch. I 1, ἢ. 40). Quite
similarly, Sext. Emp. Math. 10.278 has τὸ σημεῖον κατὰ τὸν τῆς μονάδος λόγον τετάχθαι.
106 Fr, 4 Lang = Th. ar. 82. τοῦ (= DK 44A13, KR 319); the sentence cited is at Th.
ar. 84.10f. KR, p. 255, have “little doubt” that Speusippus is reproducing genuine and
early Pythagorean doctrine, and Guthrie agrees (1 260f). Saffrey would prefer to brand the
fragment a neo-Pythagorean forgery, but there are no arguments for this—-not even the
absence of the title from the list of Speusippus’ writings in D.L. 4.4, for this is expressly
labeled as incomplete.
107 6sf.
108 Cf, above, n. 62.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
cognition, which in Platonism is sct up as parallel to the hierarchy
of geometrical solids, and also with the numbers from I to 4 (1 νοῦς,
2 ἐπιστήμη, 3 δόξα, 4 αἴσθησις),199 likewise makes its appearance as the
teaching of Pythagoras;"° the Platonic system, in all its aspects, is
being passed off as Pythagorean. Here too, along with partial agree-
ment, we find a contradiction to the statement of Aristotle that the
Pythagoreans equated δόξα with the number 2.1" The relationship
between Xenocrates and the series of types of cognition, and a certain
kinship between them and the Timaeus, have already been mentioned.!”
But we must go over the same ground once more. “Pythagoras”’
taught, according to Aétius,4* that earth came to be from the cube,
fire from the pyramid (or tetrahedron), air from the octahedron, water
from the icosahedron, and finally, from the dodecahedron, the “sphere
of the All.” The agreement with the Timaeus, extending to actual
verbal coincidence, is obvious." Less striking, but no less clear, is the
contradiction of what Aristotle says about the Pythagoreans: “They
have not said anything at all about fire or earth or the other material
things of this sort, because, I suppose, they did not have anything
distinctive to say about perceptible things.’’445 Here we may add an item
109 Above, ch. I 1, nn. 48-$4, :
110 Aét. 1.3.8; cf. 4.4.5 (Theodoret 5.20), Theo Sm. 97.24ff (the eighth “‘tetractys’’),
ps.-Archytas p. 38.12 Thesleff = lam. Comm. math. sc. 35.27ff From the fact that the
explanations in Aristotle (De an. 404b21ff) and in Aétius are different, Kucharski wishes
to conclude that they have a common source. In fact, Aétius is hardly likely to be depen-
dent on De anima, but the common source is not therefore a Pythagorean writing, but the
exposition of Platonists.
111 Alex, Met. 39.13ff = Arist. fr. 203: νοῦν δὲ καὶ οὐσίαν ἔλεγον τὸ ἕν... δόξαν δὲ
τὰ δύο διὰ τὸ ἐπ’ ἄμφω μεταβλητὴν εἶναι.
112 Above, ch. I 1, nn. 52, 57.
118 Aét, 2.6.5, and Ach. Is. 6 p. 37.206 Maass, probably from Posidonius. Cf. the
further material collected by Sachs off.—Sext. Emp. Math. 10.283, Hypomn. 25, and Anon.
Phot. 439b17 only enumerate the elements, without mentioning the regular solids—
doubtless the result of abbreviation in the excerpting process. On Procl. In Eucl. 65.15,
which is usually attributed to Eudemus (fr. 133 W.), see below, ch. VI 1.
1144 On the dodecahedron, Tim. 550: ἐπὶ τὸ πᾶν ὁ θεὸς αὐτῇ κατεχρήσατο. Aét. 2.6.5:
ἐκ δὲ τοῦ δωδεκαέδρου τὴν τοῦ παντὸς σφαῖραν. On the problem of the fifth element cf.
Guthrie I 270ff, Ρ. Moraux, RE XXIV 1171-1263.
15 Met. 990a16. Because the Pythagoreans do not distinguish between the perceptible
and the mathematical, in Aristotle’s view they are incapable of explaining the peculiar
characteristics of the physical world. The ironic explanation, in the sentence quoted, is
Aristotle’s, and this in itself shows that the Pythagoreans did not have a doctrine about the
elements. The Timaeus attempts to explain the properties of the elements (the sharp,
pointed pyramid is fire, the stable cube is earth, the nearly spherical icosahedron is water,
and so on). Raven (PylE 42 Ὁ) tries to neutralize Aristotle's evidence with Cael. 293a30ff,
where fire is characterized as τιμεώτατον, Which he thinks is an indication that there did
after all exist a Pythagorean doctrine of the clements. Aside from the fact, however, that
such a value judgment is nota “theory” in the sense required by the search for ἀρχαί,
Aristotle separates this passage from: the preceding Pythagorean material with the words
πολλοῖς δ᾽ ἂν καὶ ἑτέρυιν cuvdd€erer, Ele means the Platoniste (cl below, eh. TV a, ἢ, 16).
3. The Later Tradition and Its Sources
from the scholia to Euclid, possibly from Eudemus, whose significance
was seen by Eva Sachs.!6 The Pythagoreans, it says, treated of only
three regular solids—pyramid, cube, and dodecahedron, and Theaetetus
was the first to add the octahedron and icosahedron. But again Speusip-
pus, contra Aristotle, treated the derivation of the elements from the
regular solids as Pythagorean; he wrote in his book On Pythagorean
Numbers, ‘‘. . . and about the five shapes, which are given to the cosmic
elements” (Th. ar. 82.17 DeFalco).
The keystone of the argument is the declaration of Aétius, “ Pythago-
ras says that the cosmos is a created thing (γενητόν) in the realm of
thought but not in that of time (κατ᾽ ἐπίνοιαν... οὐ κατὰ χρόνον) .᾿᾽.11
“Pythagoras” is here taking a position in the controversy which
developed over interpretation of the Timaeus, whether the creation
of the world by the Demiurge is to be understood literally. Speusippus
and Xenocrates asserted that it was merely a mode of exposition
διδασκαλίας χάριν, an essay in mental construction (ἐπινοίᾳ).118 It is this
that gives point to Aristotle’s polemic, “There ought to be no difference
of opinion as to whether or not the Pythagoreans posit generation
(γένεσις) or not; for they say clearly...”4° Plainly, Speusippus and
Xenocrates gave out their interpretation of the Timaeus as Pythagorean
doctrine, and Aristotle is refuting them with the actual words of a
Pythagorean document. Aristotle already presupposes and rejects the
conception of Pythagoreanism that makes its appearance in the later
doxographical tradition. It is that of Speusippus, Xenocrates, and
Heraclides, which amalgamates interpretation of the Timaeus with the
Platonic “‘system of derivation.”
There remain to be discussed a number of reports in the doxography,
most of them concerning less vital questions, that cannot be placed in
direct connection with the evidence of Aristotle; the latter did not have
occasion, of course, to mention every detail of Pythagorean teaching.
Hor these additional testimonies too, however, it can be shown that
they go back to Plato’s disciples, or, insofar as they preserve older
material, were transmitted by them.
6 Schol. Eucl. 654.3; cf. below, ch. VI 3 (not in DK); the Suda s.v. Theaetetus;
Sachs 76ff. Timpanaro Cardini’s allegation (31) that Sachs brought forward against the
Pythagorean tradition “nient’ altro che affermazioni arbitrarie’’ is astonishing.—Diels
included Aét. 2.6.§ among the Philolaus materials, as 44Ar§, comparing B12; contra,
Sachs 43ff; cf. below, ch. ΠῚ 2, n. 183.
7 Aét. 2.4.1; cf. Sext. Emp. Math, 10.255: κατ᾽ ἐπίνοιαν...
ΠΡ Arist. Cael. 279b32, with the ancient comments and scholia, assembled by Lang as
Speusippus fr. $4; Xenocrates fre. 33, $4 Π.
We Met, rogtalgi; above, ch. Pa, τὸν 41
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
The kernel of Pythagorean wisdom is the “‘tetractys,” or “‘four-
group, made up of the numbers, 1, 2, 3, 4, which add up to 10. They
are represented in a pebble figure, in the form of the “perfect triangle;”’
and the available sources, from Posidonius on, show how these four
numbers contain not only the basic intervals—fourth, fifth, octave,
and double octave—but also, according to the Platonic pattern, point,
line, plane, and solid.2° The harmonic ratios, the “perfection” of 10,
and the role of the pebble figures are all part of what Aristotle attributes
to the Pythagoreans; but the derivation of spatial shapes is in implicit
contradiction to what he says.!! Speusippus’ book On Pythagorean
Numbers, on the other hand, presented this series; and half of his
presentation was devoted to the number 10, with emphasis on the
asseveration which recurs again and again in the literature on the
tetractys, that all people, Greeks and barbarians alike, count to 10
and then return to 1.1%? But there is a relation to Xenocrates as well.
It is frequently mentioned that the Pythagoreans, in their oath by the
tetractys called it the “fount of ever flowing nature (παγὰν ἀενάου
dvcews). 5 But Xenocrates called matter dévaos,!24 and, though this
may also hold an allusion to an expression of Plato’s, it is tempting
to believe that the verse on the tetractys was known to him.!%5 For
120 Sext. Emp. Math. 7.94-100, with the parallels (above, nn. 8, 12~14), Aét. 1.3.8,
Theo Sm. 87.5ff, 93.17ff, etc. The most detailed treatment is that of Delatte, Litt. 249-
268. Sometimes it is also called τέταρτος ἀριθμός (cf. Theiler, Gnomon 28 [1956] 284f.)
Lucian (V. auct. 4) represents Pythagoras as saying, ‘‘ What you suppose is four is really ten,
and a perfect triangle, and our Oath.” Cf. also Burnet, EGP 102f; below, ch. II 4.
221 Above, n. 92.
122 Th. ar. 83.7: ὀρθῶς τε καὶ κατὰ φύσιν els τοῦτον καταντῶμεν παντοίως ἀριθμοῦντες
“Ελληνές τε Ket πάντες ἄνθρωποι, (Arist.) Pr. 15.3.910b23ff: διὰ τί πάντες ἄνθρωποι,
καὶ βάρβαροι καὶ Ἕλληνες, εἰς τὰ δέκα καταριθμοῦσι.... εἶτα πάλιν ἀναδιπλοῦσιν;
Sext. Emp. Math. 4.3: ἐπ᾽ αὐτὸν φθάσαντες πάλιν ἀναλύομεν ἐπὶ τὴν μονάδα (at 7.94 the corre-
sponding sentence has dropped out); Philo Op. 47: περὶ ὃν ὡς καμπτῆρα εἱλοῦνται καὶ
ἀνακάμπτουσι. Quite similar is Anat. p. 39 = Th. ar. 86.3; cf. Theo Sm. 93.19, 99.17,
Anon. Phot. 43925, Hippol. Ref. 6.23.3. There is a remarkably close resemblance between
the passage from the Problemata and the following of Aét. 1.3.8: μέχρι yap τῶν δέκα πάντες
“Ἕλληνες, πάντες βάρβαροι ἀριθμοῦσιν, ἐφ᾽ ἃ ἔλθοντες πάλιν ἀναποδοῦσιν ἐπὶ τὴν μονάδα.
The common source is obviously Speusippus.
123 Below, ch. IE 4.
MeRr 2H. || Abt 1.3.23.
™ For this combination, see Zeller ΠΠ itorg, n. 4; Cherniss, Plato 484}, Guomon 1ysy,
41; against it, Heinze 14. Delatte is dubious, Lith 241, 0]. Leg. φόρος. --cf also the
doxographical reports on Pythagoras’ teaching about ὕλη ῥευστή; above, n. 45.
3. The Later Tradition and Its Sources
another verse, too, is commonly cited in the literature along with this
one: ἀριθμῷ δέ τε πάντ᾽ ἐπέοικεν. and Themistius cites this very line
in a passage where he refers to Xenocrates’ book On Nature.!2? Though
a good many bits of Themistius’ own wording may have worked their
way in, alongside the paraphrase of Aristotle, still he was well informed
about Xenocrates.1?8 |
Thus from an element of Pythagorean tradition which at first
seems to contain an ancient, pre-Platonic kernel, a clear trail leads us
back to Speusippus, and a somewhat less clear one to Xenocrates.
It is the latter to whom the connection seems closer in the matter of the
daimones. It is quite probable that Pythagoras himself spoke of δαίμονες,
but the process of transition from religious experience and teaching to
systematic philosophy is obscure. In any case, the expressions in the
later tradition, about daimones as souls or as some kind of intermediate
being, were the work of Platonists. Aétius (1.8.2) names Thales,
Pythagoras, Plato, and the Stoics as advocates of doctrines about
daimones. Plutarch is more cautious: in two parallel passages he names,
first, ““Plato, Pythagoras, Xenocrates, and Chrysippus”’ (De Is. et Os.
25.360d), and then again only “Plato, Xenocrates, and Chrysippus”
(De def. or. 17.419a). That Pythagoras is not named first, in despite
of chronology, or even may be left out, suggests the conjecture that
what is meant is really “Pythagoras as cited by Xenocrates.”’!29 If,
as Aristotle says (fr. 193), the Pythagoreans considered it the most
natural thing in the world for a person to “see” a daimon, and if they
also saw the myriads of souls dancing as motes in a sunbeam,}*° this is
a long way from the Platonic concept of immateriality. There is also
the graded series god-daimon-hero-man, which was attributed to the
126 Sext. Emp. Math. 7.94, 4.2, Theo Sm. 99.16, Plut. De an. procr. 1029f, Iam. VP
162, etc. Delatte, Litt. 14f, would like to show that the line is from an ancient hexameter
‘lepos λόγος, but the pareomiac (with a catachrestic dependence on 1]. 22.71 and Tyrtaeus
7.27) may be independent.
127 Xenocrates fr. 39 H. = Themist. an. 11.20ff.
28 Themistius knew this book of Xenocrates, as is shown by fr. 61 Heinze (Cherniss,
Plato 399 n. 325, vs. Heinze p. 65). Saffrey (38ff) would like to posit a “neo-Pythagorean
revision” of the physics of Xenocrates, for to him ‘‘Pythagorean” means “non-Xeno-
cratic.” But the connections shown here leave no room for doubt; Xenocrates himself
Δ Pythagorizes.”
10 Heinze 78ff. M. Detienne seeks to recover original Pythagorean material: REA 60
(1948) 271-279; RHR τὲς (1959) 17-32; Homere, Hésiode et Pythagore (Brussels, 1962);
La notion de Datmon dans le pythagorisme ancien (Paris, 1963); on the last, Burkert, Gromon
16 (1964) 563-567; cf. below, ch. If 2.
0 De an. 404a17fF, “a piece of Pythagorean superstition,” says Zeller 1 $61, and Cher-
niss agrees (Pres. 291 n. 6). But cf. Democr. ap. Arist. De an. 404a3ff() DK 67A28). The
Hypomuemata have here preserved ancient material (32): εἶναί re πάντα τὸν ἀέρα ψυχῶν
ὄμπλεων.
I PLATONIC AND PYTHAGOREAN NUMBER THEORY
Pythagoreans from the time of Aristoxenus.’* There is no direct way
of checking this; nevertheless, this hierarchical series is common in
Plato2%2 while Aristotle ascribes to the Pythagoreans a three-term
series, with Pythagoras himself as middle member between god and
man (fr. 192).
In addition to such beliefs about souls and daimones, the doxographical
tradition assigns to Pythagoras or the Pythagoreans the threefold
division of the soul into νοῦς, θυμός, ἐπιθυμία,138 and the authority of
Posidonius is cited for this.% But it is apparent by now that this is
no isolated phenomenon; in other respects as well Posidonius appears
as a link in the Pythagoras tradition. Now, this tripartition of the
soul is closely related to the theory of the “three ways of life,” which
Heraclides Ponticus cites as coming from Pythagoras.1% It is possible
that the soul doctrine insinuated its way into the tradition by the same
route. Anyway, here too Platonic and Pythagorean material is seen
as a unity, and Aristotle’s reports lead in a different direction.”
The Hypomnemata present a somewhat different division of the psyche
(30) “into νοῦς and φρένες and θυμός. νοῦς and θυμός are also present
in the other animals, but φρένες only in human beings.” Von Fritz
has tried to show, on the basis of a thorough semantic investigation,
that in this respect the author is reproducing a completely un-Platonic
and therefore pre-Platonic doctrine. 138 From Empedocles on, φρονεῖν
and νοεῖν were generally equated, and in all post-Platonic philosophy
νοῦς is a specific criterion of humanity; therefore a theory which
181 DL. 8.23, Iam. VP 100, 144, 175, Por. VP 38; cf. Aristox. fr. 34, lam. VP 37,
Zaleucus ap. Stob. 4.2.19, Carm. aur. 1-4.
132 P], Crat. 397¢-e, Rep. 392a, 427b, Leg. 717b, 738d, 801e, 818c, 9108, Epin. 984d.—
Inscriptions from Dodona (Collitz II 1582, 1566, 158 sb), apparently from Epirus, Corcyra,
and Aetolia, point toward a ritual background.
188 Posidonius in Galen Plac. Hipp. et Plat. 4.7, 5.6 (V 425, 478 K.), with the cautious
formulation αὐτοῦ μὲν τοῦ Πυθαγόρου συγγράμματος οὐδενὸς εἰς ἡμᾶς διασῳζομένου,
τεκμαιρόμενος δὲ ἐξ ὧν ἔνιοι τῶν μαθητῶν αὐτοῦ γεγράφασιν. Aét. 4.4.1, cf. 4.7.5 (“ Pythag-
oras and Plato”). It is common in the pseudo-Pythagorean literature: Archytas fr.
64 Nolle = Iam. ap. Stob. 1.49.34; Aesara (or Aresas) ap. Stob. 1.49.27 = Thesleff,
Texts pp. 48ff; Metopus ap. Stob. 3.1.115 = Thesleff, Texts p. 116; Theages ap. Stob.
3.1.117 = Thesleff, Texts p. 190. Cf. also Pollux 2.226.
134 For this reason the tradition is often defended: Burnet, EGP 296 n. 2, Taylor,
Tim. 263f, 496ff, Cornford, CQ 1922, 147. Joly hesitates, pp. 78f, with refs. Philolaus fr.
13 has nothing to do with this (below ch. ΠῚ 2, nn. 149-157).
135 Above, nn. 8-22, 74-75.
189 Err, 87-88 Wehrli; cf Burkert, Hermes 1960.
87 Not only the motes in the sunlight, but also the allusion to metempsychosis at
De an. 407b22 (below, ch. Π 3), On ψυχή and ἁρμονία, see below, ch. ΠῚ 2, nn, 165-168.
18 Node, νοεῖν, and Their Derivatives in Pre-Socratic Philosophy,” CP 41 (1946) 33f;
similarly Wellmann, ilermes τι, 235th; Delatte, Mie 222.
3. The Later Tradition and Its Sources
assigns vods to animals as well must go back to the first half of the
fifth century B.c. Also, φρένες, he says, was not used from the fourth
century on.
Still, φρένες does occasionally appear in later writings,!8® and it
remains arguable that at least a certain kind of vods belongs to animals.1#°
More important, however: the evidence of the Hypomnemata is not
isolated. Aétius tells us (5.20.4), “Pythagoras and Plato say that the
souls of the so-called irrational animals have reason too, but they do
not act in accordance with reason because of the faulty composition
[mixture] of their bodies, and because they do not have the faculty
of speech (τὸ φραστικόν). We can see this in the case of apes and dogs;
for they think (νοοῦσι) but do not speak (dpdfover).”” What Aétius
says about thinking (νοεῖν) and speaking (φράζειν) now makes clear
the distinction between νοῦς and φρένες in the Hypomnemata. There are
other ways, too, in which the two sources resemble one another in
their treatment of “Pythagorean”’ psychology.™! In addition, Aétius
speaks of Pythagoras and Plato in a single breath, in this specific
context, so that we cannot single out particular un-Platonic or pre-
Platonic elements. Here, too, we have an interpretation of Plato
that, like Aristotle, restricts immortality to the highest psychic activi-
ties, and therefore divides the intellectual segment of the psyche in
the same way as the Stoic differentiation between λόγος ἐνδιάθετος
and λόγος προφορικός.145 This is not a trace of pre-Platonic Pythagorean-
ism.
There remain two ancient-sounding reports of Aétius on doctrines
of Pythagoras. The first says that necessity hedges the world about:
ἀνάγκην περικεῖσθαι τῷ κόσμῳ,133 and the second is a definition of time
139 Cf. Festugitre REG 1945, 43ff; e.g., Chrysippus, SVF IH p. 27.8; Anon. Lond.
4.14ff, Phid. Po. 5.19, Rhet. 1.240 5. When the “diaphragm” is misplaced in the head
(Hypomn. 30), we scarcely get the impression of having the ancient or original version
(“‘correctly” Hebd. 6: moon—¢p7v—mid-point—seat of intelligence).
140 According to Aét. 4.5.12, Democritus followed Parmenides and Empedocles in
equating νοῦς and ψυχή (cf. Arist. De an. 404a27ff). Strato (fr. 107ff Wehrli) equates νοῦς
and αἴσθησις; in such circumstances one could scarcely deny the lower animals any
share in νοῦς.
141 Aét. 4.5.10: Πυθαγόρας τὸ μὲν ζωτικὸν περὶ τὴν καρδίαν, τὸ δὲ λογικὸν καὶ νοερὸν
περὶ τὴν κεφαλήν. Cf. Ηγρονπη. 30: νοῦς and φρένες in the brain, θυμός in the heart.—Aét.
4.7.5: Πυθαγόρας Πλάτων τὸ μὲν λογικὸν ἄφθαρτον, τὸ δὲ ἄλογον φθαρτόν. Cf. Hypomn.
10: τὸ μὲν φρόνιμον ἀθάνατον τὰ δὲ λοιπὰ θνητά.
142 Aétius cannot be drawing from the Hypomnemata, since he would not have found the
references to Plato there. The Hyponimemata are probably older than the Vetusta placita
(cf. the Stoic δυσκρασία, Att. §.20.4).
143 Att, 1.25.2; Zeller (I $42 n. 2) is inclined to equate Necessity with the fire that
surrounds the world.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
as “the sphere of the encompassing (σφαῖρα τοῦ mepiéxovros).” This
seems to be known to Aristotle, who refers to it in connection with the
definition of time abstracted from the Timaeus!4* and dismisses its
reasoning, ‘“‘that everything is in time and likewise in the sphere of
the all” as “‘somewhat simple-minded.” Indeed the association of the
celestial sphere and all-embracing time was widespread in the fifth
century'® and evidently reflects a pre-philosophic way of thinking.
It makes its way into Pythagorean doctrine as attested by Aristotle:
the cosmos breathes in time, along with the Unlimited; it is still
discernible in Aristotle’s allusion to the divine aion encompassing the
world,!*” though transformed by Plato’s distinction between aion and
chronos (Tim. 37d). Later tradition points to Orpheus for this imagina-
tive concept: Chronos hatching the world egg.'#8 And Ananke, too,
occurs in Orphic mythology, expressly associated with Chronos
surrounding the world.!*° It is possible that Orphic cosmogonies were
exercising some influence as early as the fifth century and that they
were in closest connection with Pythagoreanism. The way by which
144 Aét, 1.21.1, Arist. Phys. 21833 (τὸν χρόνον) of μὲν yap τὴν τοῦ ὅλου κίνησιν εἶναί
φασιν, of δὲ τὴν σφαῖραν αὐτήν. For the attribution of the first definition to Plato (Tim.
37c-d, 39b-c), cf. 223b21, Theophr. Phys. op. fr. 15 (Dox. 492), Eudemus fr. 82 W., Aét.
1.22.1, Plot. 3.7.8. Simplicius (Phys. 700.16ff, 785.14) thinks the definition of “ Pythagoras”
is to be found in the Categories of the ps.-Archytas (24,15 Thesleff) (χρόνος) κινάσιός
τινος ἀριθμὸς ἢ καὶ καθόλω διάστημα τᾶς τῶ παντὸς φύσιος. Speusippus (fr. 53 L.) and
Xenocrates (fr. 40 H.) have different definitions of time.—According to Plutarch (Quaest.
Plat. 1007b) Pythagoras defined time as ὅλου ψυχή. (Cf. the ps.-Archytas definition of
time with Speusippus’ definition of soul, above, n. 74.)
145 Hermippus the comic poet, fr. 4 (= Stob. 1.8.36; ca. 430-420 B.C.):
, a? 3 f a ¥ 4 é
ἐκεῖνός ἐστι στρογγύλος τὴν ὄψιν, ὦ πονηρέ,
ἐντὸς δ᾽ ἔχων περιέρχεται κύκλῳ τὰ πάντ᾽ ἐν αὐτῷ....
> é 3} fa Ἄ i 4 4
4 ὀνομάζεται δ᾽ ἐνιαυτός, ὧν δὲ περιφερὴς τελευτὴν
οὐδεμίαν οὐδ᾽ ἀρχὴν exer...
Cf. Hebd. 16: ‘annus autem, in quo omnia circumeunt, habet in se ipso omnia,”’—The
heaven is χρόνου ποίκιλμα in Critias B18 (cf. B25, 33f). In Indian mythology too, Kala or
Time is described as being the starry heaven (Olerud 135).
146 Arist. fr. 201 (cf. Cael. 279a11). This is not identical with the Pythagoras testimony
Act. 1.21.1: according to these Pythagoreans, the sphere of the universe came to be, but
nobody spoke of coming-to-be of time before Plato (Arist. Phys. 251b17); and the role
of the celestial sphere for measuring time was to change radically through the introduction
of a moving carth (ch. ΓΝ 3).
147 Cael. 279018-b3.
M8 Kern, Orph. frag. $4, $7, 70.
1 Kern, Orph. frag. $4, 126, 162; Nicomachus ‘Uh. ar, 81.19: τὴν ᾿Ανάγκην ot θεολόγοι
τῇ τοῦ παντὸς οὐρανοῦ ἐξωτάτῃ ἄντυγι ἐπηχοῦσι. The Ananke of Parmenides (B 8.30,
10.6) is assigned to the mid-point of the universe (A37). On Orphism and Pythagoreans,
above, ch. 12, nn. §3-§6; below, ch. ἢ.
3. The Later Tradition and Its Sources
these two isolated statements came to be handed down in the doxo-
graphy as lore of ‘‘Pythagoras” is not clear; but there is one obvious
guess: the Plato interpretation of the Old Academy played a decisive
role. For as the definition of time (chronos) is treated by Aristotle in the
context of exegesis of the Timaeus, so ananke belongs to the Republic,
where the universe is whirling on the knees of Ananke (616) 15
Finally, there is the basically important doxographical statement
that Pythagoras coined the term “cosmos, 151 ἐκ τῆς ἐν αὐτῷ τάξεως
(Aétius). This makes him the creator of one of the most influential
concepts of Greek natural philosophy. The history of the word in
pre-Socratic philosophy, as far as we can trace it, speaks, however,
decisively against the thesis of such a programmatic invention of the
concept at the very beginning of Greek philosophy; we rather sce
the word gradually developing, from the prosaic “arrangement” of
things in parts of the universe to the “ordering” of the whole world.
There is, nevertheless, again an obvious guess how the alleged invention
came to be attributed to Pythagoras: it fits very well into the anecdote
of Heraclides Ponticus which had Pythagoras explaining the word
philosophia—an invention of Pythagoras which also made its way into
all the doxographical handbooks.'*
Nevertheless, it is communis opinio that the Pythagoreans at least,
if not Pythagoras himself, played a decisive role in the development of
the Greek idea of “cosmos.’’54 One always cites Plato, who introduces,
against Callicles, the doctrine of “wise men” that friendship and
equality hold sway in the world, and that this is why it is called
180 Clearchus (frr. 3-4 Wehrli) wrote a commentary on the mathematical passages of
the Republic.
181 A&t, 2.1.1 = DK 14.21 (τὴν τῶν ὅλων περιοχήν), D.L. 8.48 (τὸν οὐρανόν; the additional
notice about Parmenides relates only to the spherical shape of the earth, as Diels saw,
Dox. 492); Anon. Phot. 440a27f (τὸν οὐρανόν); Achilles Is. § (τὸ πᾶν); Schol. A Il. 3.1
(ἡ τῶν ὅλων τάξις); lam. VP 162. On the cosmology attributed to “Pythagoras” in the
doxography, below, ch. IV 1.
152 Cf, J. Kerschensteiner, Kosmos: Quellenkritische Untersuchungen zur vorsokratischen
Philosophie (Munich, 1962); H. Diller, “Der vorsokratische Gebrauch von κόσμος und
κοσμεῖν, in Festschrift Snell (Munich, 1956) pp. 47-60; Kirk, Heraclitus 311ff (only the
sense ‘arrangement of things” in Heraclitus; contra, Dodds, Gorgias 338f, Kerschensteiner
y7ff).—Kranz, ABG 1955, 13f, traces the word κόσμος back to Anaximander, as do Diels,
Jaeger, Gigon (Ursprung 91), Hélscher (Hermes 1953, 264), and Kahn (219ff). Kerschen-
steiner (208) qualifics this.
188 Sq Kirk, Heraclitus 313.1, Kerschensteiner 229; cf. above, n. 77, and below, ch.
IV τ.
4M. Pohlenz, Aus Plates Werdezeit (Berlin, 1913) τ 528, Kranz, Philologus 1938, 439,
ABG 1955, 35, DK Nachtrige 1 502.17; Dodds, Gorgias 337ff; Krimer 142, 233; Guthrie
1 206ff.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
“cosmos.”’155 Here Plato adduces the evidence of geometry, in a manner
that cannot be well explained from the dialogue itself, and there is
much to be said for the conjecture that Plato is alluding to the close
affinity between geometry and the study of first principles which he
had learned from Archytas.1® It is also possible, however, that he was
already foreshadowing ideas of his own, later formulated in the Ti-
maeus; and the close relationship of this passage of the Gorgias to other
thinkers of the fifth century, especially Empedocles and Euripides,
ought not to be overlooked because of concentration on the Pythago-
rean allusions.15’ In spite of the distinctly Pythagorean flavor of the
155 Gorgias §07e—508a: ‘‘ Wise men say, Callicles, that heaven and earth, gods and men,
are held together by the principles of sharing, by friendship and order, by self-control and
justice; that, my friend, is the reason they call the universe ‘cosmos,’ and not disorder
or licentiousness. Clever though you are, you seem not to have paid enough attention to
these matters; it has, in fact, escaped you what a mighty power is exercised, both among
men and gods, by geometrical equality. And it is your neglect of geometry which brings
about your opinion that one should strive for a share larger than that which other men
possess”’ (tr. W. C. Helmbold).
156 Cf. above, nn. 98f. Kerschensteiner 222f, Dodds, Gorgias 339f, and others under-
stand ἡ ἰσότης ἡ yewperpixy in the sense of the “geometrical proportion” that was
considered the principle of aristocratic justice, in contrast with arithmetic equality, the
principle of democracy: Pl. Pol. 2570, Arist. EN 1131b13ff, Pol. r301b20ff; cf. Pl. Rep.
558c, Leg. 757bc, Isoc. 7.21. Ps.-Archytas 34.3-14 Thesleff has a different scheme, in which
the harmonic mean is aristocratic, the geometric democratic, the arithmetic oligarchic.
(Cf. Delatte, Pol. 95ff, with further references; the spuriousness of the Archytas material
is proven even by the use of such a word as ἀπάθεια, 33.18 Thesleff). But an “equality”’
that distributes “‘more to the greater and less to the lesser” (Pl. Leg. 7570) would be
scarcely appropriate to refute the pleonexia of Callicles; and “geometrical proportion”
usually is γεωμετρικὴ ἀναλογία, or γεωμετρικὴ μεσότης, not simply γεῳμετρικὴ ἰσότης.
Thus “geometrical equality” should be understood in a more general sense, as ἡ τοῦ
ἴσου ἀναλογία in Archytas A23a—the power of mathematics that governs the world.—
Plato, in comparing justice to musical harmony, and therefore to the “harmonic’”’ mean
(Rep. 443d; Morrison, CQ 1958, 213ff), shows that he is not committed to any specific
pattern.
157 Olympiodorus, on the passage, and the scholiast, name “the Pythagoreans and in
particular Empedocles.” And, to be sure, Φιλότης is a cosmic power in Empedocles,
“equal in length and breadth” (B17.20); mathematical proportion plays a role (B96, cf.
135). Most similar to the Gorgias passage is the exhortation of Jocasta in Eur. Phoen. 53 5ff
about the “equality” that binds man to man, as it governs the course of days and years.
The same kind of protreptic is presupposed by Soph. Aj. 67o0ff. But cf. also (after Diog.
Apollon.), Hippoc. Flat. ς (DK 64Cz) and Pl. Crat. 412d, on the d:efidv.—A reference to
Pythagorcans is seen in the word σοφοί (Dodds, Gorgias 297), in comparison with Meno
81a, Rep. §83b (Adam II 379, ad loc.), and Gorg. 493a (below, ch. II] 2,n. 48). But the word
is much more widely used. Plato seems to apply the word σοφός to Empedocles (Lys.
214b), to Heracliteans (Lys. 215d), and to Sophistic rhetors (Lys. 222¢, Meno 75c, Phdr.
266c. Symp, 18$c). Cf. also Crat. 4020 (Heraclitus), Phdr. 235¢ (Anacreon), Rep. 36$¢
(Simonides), ‘Tht, 157b (Heracliteans?). RS. Bluck, too, doubts that σοφοί is used here of
the Pythagoreans (CR τὰ [{τῶ61] 30). Phe ancient tradition connects with Anaxagoras
the well-known verses of Euripides which say that the contemplation of the “unchanging
cosmos” restrains from evil deeds (Hur. fr. gto - Anaxagoras Ajo); and even as carly
as Anaximander (fr. 1) there is a justice involved in cosmic process,
3. The Later Tradition and Its Sources
passage, the mention of geometry certainly does not point back to the
oldest stratum of Pythagorean mathematics, in which arithmetic was
central.5®8 Thus in no instance does the doxography on Pythagoras
the philosopher bring us back to solid ground; it all seems to derive
from the Platonic exegesis of the Old Academy.
Zeller’s solution of the problem of the Pythagorean tradition must,
then, be modified. It is not true that we can make a chronological
distinction between the oldest and therefore most reliable accounts,
those of Aristotle, and a later accretion which cannot be checked and
is therefore suspect; nor is it true that neo-Pythagoreanism, no matter
how early one makes it begin, is to be held accountable for the changing
course of the tradition. In the earliest evidence available two concep-
tions of Pythagorean philosophy are in sharp contrast with each other.
The non-Aristotelian tradition is even the older of the two, insofar as it
can be traced to Speusippus. This tradition from the Old Academy
equates the Pythagorean philosophy with the doctrines of the Timaeus
and with the Platonic number philosophy. The highest principles,
immaterial, are the One and the Indefinite Dyad; from them come the
pure, incorporeal numbers; the numbers produce the pure geometrical
shapes, line, plane, and solid, as well as the perceptual functions of the
“Living Creature itself’; from the mathematical regularity of the
regular polyhedra come the elements and therewith the multifariousness
of the empirical world; and this whole process takes place, in this
order, only in thought, which traces back changeless Being to its
ultimate principles and understands it by means of them.
In each of these points the exposition of Aristotle offers contradiction.
The Pythagoreans do not think of separating the numbers from the
physical world, but stay in the realm of the perceptible; the numbers
are spatially extended shapes, and things “are” numbers; order,
correspondence, harmony within the empirical world are com-
prehensible in terms of number, which is the key to understanding and
shows forth its power in divers ways. The world, which is single, has
come to be, as harmony, out of Limit and the Unlimited, and its
articulation expresses the perfection of number. There are hints of
mythical ways of thought, and obviously there is little logical or
systematic consistency.
That in fact the interpretation of the Old Academy does form part
of the background for Aristotle is shown by his clearly polemical form
188 Below, ch. VI 2.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
of expression: there ought to be no difference of opinion as to whether
or not they attributed generation to the cosmic numbers.!* Aristotle
at times, to be sure, shows that he has roots in the other tradition.
In the De caelo he joins in the Pythagorean number juggling,!® in his
On Philosophy he follows right along in the Pythagorean tracks in
explaining the perfection of musical harmony, taking his departure
from ἄπειρος and περαίνουσα φύσις 161 in the first book of the Meta-
physics he treats Plato and Pythagoreans as quite closely connected,
whereas in books M and N he scarcely mentions anything but their
disagreements.1®
In consideration of these matters, the question of a possible Platonic
“early period” in Aristotle’s thought is bound to rise. The De caelo
and the first book of the Metaphysics are thought to be early works ;1¢
it is possible that in his understanding of the Pythagorean philosophy,
as well as in other matters, Aristotle only freed himself gradually from
the picture of it drawn in the Old Academy. But, given the compli-
cated and controversial character of the problem of Aristotle’s philoso-
phical development, this cannot be anything but a conjecture. It may
help us, however to understand the surprising fragment 207, without
taking refuge in athetesis: “In his writings on Archytas, Aristotle
says that Pythagoras too called matter ‘other, because it is in flux
(ῥευστήν) and always becoming ‘other’ than it was.’”!*4 This obviously
has connections with the “ matter in flux (ὕλη pevory)”” which according
159 Mfet. 1091aI 3ff.
160 Cael. 268aI0.
161 Fr, 47 = Plut. Mus. 24.1139f; attribution to On Philosophy (fr. 295 Walzer) by
E. Bignone, Ann. d. Scuola Norm. Sup. d. Pisa Ser. II 2 (1933) 287, L’Aristotele perduto II
(Florence, 1936) 361f. Mondolfo (ΖΜ 373) thought it might be attributed to the book
on the Pythagoreans, but he can scarcely be right, since that book contained criticism as
well as exposition (fr. 205). Ross (Aristotelis fragmenta selecta, p. 93) accepted the false
conjecture ἀρτίου καὶ περισσοῦ for ἀπείρου καὶ περαινούσης .. . φύσεως; cf. below, ch. III
2.
162 We cannot here go into the question of the attempt of Cherniss to reconstruct a
common original for Metaphysics A and M (Plato 189ff).
163 Jaeger, Arist. 175}, 316ff. For a brief survey of the later research, cf. De Vogel
GP II 10-19.I. Diiring, Aristoteles (1966) 254-270, assigns A and MN to about the same
period.
164 Fr. 207 = Damasc. Princ. 2.172.16ff. Because of its inconsistency with Aristotle’s
other statements, the authenticity of this fragment was denied by Zeller I 470 n. 3,
Cherniss, Pres. 17 ἢ. 68, De Vogel, Pythagoras 21 4ff. Rostagni, whose reconstructions of
Pythagoreanisin follow Platonic paths, defended its genuineness (Verbo 43.1), citing Met.
1087b26, whose contrast of “other” and “one” is attributed by ps.-Alexander 798.23
to Pythagoreans (cf. Ross, Mef. Π 471). But, since the question is that of the characteriza-
tions of matter (ὅλη), Aristotle cannot be speaking of the Pythagoreans (cf. above, ch. I
2, 2..95). What we have is exegesis of Plato (cf esp. Pari. 1s 7b).
3. The Later Tradition and Its Sources
to the doxographers was part of the doctrine of Pythagoras and Plato.1®
If the doxography goes back, in general, to the Old Academy, it is
possible that Aristotle too, either in an early work, or in an exoteric
discussion, perhaps in a looser, dialogue form, adopted the Pythagoras
of his colleagues.16 Maybe he was attempting a philosophical inter-
pretation of a traditionary pronouncement of Pythagoras, in which the
word “other” (ἀλλο) occurred.
But if the tradition rejected by Zeller proves to be that of the Old
Academy,}® it is by no means thereby rehabilitated. It would be rash
to assume that if we merely accepted tradition the problem would be
solved. For what we have is not a unitary tradition, but irreconcilable
contradiction, and one cannot endorse one side without rejecting the
other. Scholarship must decide between the Platonic and the Aristote-
lian line, for only one of them can be historically correct.
It is natural to try to settle the matter by use of the idea of develop-
ment. The accounts of Aristotle have a decidedly antique air; they
seem “pre-Socratic, 168 while the doxography stemming from the
Platonists presupposes a sophisticated use of dialectic. But it is out of
the question to suppose that Aristotle is referring to Pythagoreans of
the fifth century, the Platonists to contemporaries. Aristotle says
explicitly that not Pythagoreans but Plato introduced the notion of the
Indefinite Dyad, and Speusippus speaks of the “‘ancients,’’ Xenocrates
and Heraclides of Pythagoras himself. For the same reason we must
reject the hypothesis, occasionally considered as a last resort, that
Pythagoreans of the fourth century could have taken over the theory
of ideas and the dialectic developed in the Academy;1®° Plato’s disciples
are not talking about younger contemporaries, but about Pythagoras.
Either Aristotle arbitrarily reinterpreted Pythagoreanism as primitive
165 Above, n. 3$.
166 Zeller explained MM 1182a11 (cf. 1194429) as a cruder version of 1078b21 with
Πυθαγόρας instead of Πυθαγόρειοι. Dirlmeier defends the authenticity of MM (Aristoteles
Werke 8, Berlin 1958), and it might well transmit early Academic tradition.
167 Zeller saw clearly the contradiction of the traditions (I 465ff), but did not come to this
conclusion, for several reasons: He did not originally perceive the dependency of the
doxography on Theophrastus, and this led him to explain away the direct testimony of
Theophrastus (above, n. 59). Also, he relied solely on the dialogues for his interpretation
of Plato, rejected the reports of Aristotle, and therefore did not see the inner connection
of the “system of derivation.”’ Finally, it is only the new fragment of Speusippus that
proves the traditions are in contradiction even in the matter of the ἀρχαΐ, and not only in
relation to line-plane-solid and the doctrine of the elements.
168 Above, ch. I 2.
169 E.g., Ross, Met. If 471 (cf. above, n. 164), Kucharski (above, ch. T 1, n. 49).
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
speculation, or the tradition of the Old Academy is, from a historian’s
point of view, fiction.”
The decision can scarcely be in doubt: Speusippus, Xenocrates, and
Heraclides equate the doctrine of their master Plato, and therewith also
their own philosophical positions, with the wisdom of Pythagoras,
whereas Aristotle sees both Academic and Pythagorean philosophy
from a certain distance, and even from the stance proper to an active
polemic.! Plato’s pupils ignore the intellectual developments of a
century and a half, a period of tremendous intellectual turmoil; their
picture of Pythagoras is, in the history of thought, an impossibility.
The consequences of this decision must be faced, without further
attempts at compromise. The “derivation system” is an achievement
of Plato and the Academy, a genuine transposition platonicienne!”® of
an older, Pythagorean number philosophy, making use of some
improvements introduced in the circle of Archytas. The tradition that
attributes the “derivation system” to Pythagoreans, or even to Pythag-
oras himself, is to be taken for what it is: evidence for the Old
Academy, for Speusippus, Xenocrates, and Heraclides, but not for
historical Pythagoreanism.!”? This involves, as mentioned above,
nearly the whole of the post-Aristotelian tradition on Pythagoreanism.
At least in the philosophical realm,! the only usable evidence for
170 The contradiction of the two traditions has mostly been overlooked, except by
Zeller; it was only the interest devoted to the Platonic number philosophy that brought
it clearly into the foreground. To be sure, people have often perceived the connection
between the later tradition and Platonism. (On Sext. Emp., above, n. 4; but Raven, CQ
1951, 147ff, saw in Sext. Emp. Math. 10.248ff at least partially Pythagorean material,
perhaps going back to Archytas. On Aétius, cf. Frank 260 n. 1.) Less clearly seen was the
contradiction with Aristotle. Delatte (Vie 234) asserts that the Hypomnemata show aucune
influence of Platonic doctrine, in spite of the Indefinite Dyad in section 25; on the other
hand Wellmann (Hermes 1919, 227) thought he could prove by Aét. 1.3.8 that the Indefi-
nite Dyad was genuine Pythagoreanism. The two testimonies stand or fall together.
(Rightly, on Hypomn. 25: Wiersma, Mnemosyne 1942, 97; Festugi¢re REG 1945, 10ff,
with refs.) Often, too, the difference is simply ignored, as by Cornford, PlParm 3ff-
171 Similarly, the Platonists always present Parmenides as simply Platonic, while
Aristotle emphasizes the Eleatics’ “primitive” inability to get beyond sense perception
(αἴσθησις) (DK 28A24-25).
172 A phrase coined by A. Diés, Autour de Platon (Paris, 1927) 400ff.
1 Throughout antiquity, and beyond, the interpretation of Pythagorean doctrines
offered by Plato and his disciples held the field; but, though the stream that was later
accepted as Pythayoreanism came from Plato, one ought not to designate Plato and hts
pupils as Pythagoreans. They are thinkers sui generis, and later “Pythagoreanism”’ is
very largely a dimidiatus Plato, This book comprehends as Pythagoreanisin that which was
not “transposed” by Plato, that which existed before Plato and contemporary with him as
the intellectual position and the way of life of persons who called themselves Pythagoreans.
(On the origins of “neo-Pythagoreanisn” see below, ch. 1.4.
"τὰ We shall see in chapters Vo and Vi that the character of che tradition ts no different
in the ficlds of mathematics and astronomy,
4. Pythagoreanism in Plato
what is Pythagorean, aside from pre-Platonic testimonies and Aristotle,
is in statements that agree with them and are not contaminated with
Platonic material.’
One might still hesitate to accept a conclusion that leads to disagree-
ment with the overwhelming majority of the sources. But it is sup-
ported by Plato himself; the significance of this evidence becomes
clear only when it is seen as foreshadowing the conflict of interpretation
which arose among Plato's successors.
4. PYTHAGOREANISM IN PLATO AND THE ORIGIN IN PLATONISM
OF THE PYTHAGOREAN TRADITION
Modern scholarship on Plato has taught us to see, in the totality
of his work and in the individual dialogues, the inner consistency and
the necessary unity. The tendency to look for concrete historical
relationships, for the origin of particular inspirations, or the target
of specific allusions, has receded into the background.! Not the least
important cause of this is a certain resignation, for it is quite impossible
to determine and delimit, from the study of Plato alone, all of his
“sources.” In Plato, every thesis or argument derives its importance
from its truth value, not from its origin in one source or another.
Foreign material is no longer foreign, but an integral part of the
Platonic structure. This is why the question of the nature of historical
Pythagoreanism is perhaps hardest of all to answer, from Plato alone.
Plato’s dialogues do not suggest strongly that Pythagoreanism
was the determinative influence upon him. Aside from Socrates, three
thinkers in particular stand out: Heraclitus, Anaxagoras, and Par-
menides.? The importance of Parmenides seems to increase in the later
dialogues: alongside the one Timaeus stand Parmenides, Sophist, and
Politicus. All the same, the few references to Pythagoreanism are of
special relevance in their Platonic context.
Pythagoras is named in a single passage in the Republic, in the
final reckoning with Homer. Has Homer, Socrates asks, earned the
laurels of a lawgiver like Lycurgus, Charondas, or Solon, or even
175 This result is decisive in the case of the Philolaus fragments (ch. III, below). To be
sure, bits of genuine Pythagorcanism make their appearance in the Platonists’ tradition,
as with the tetractys (above, n. 120). But in cach case authenticity must be proven.
1 Cf. Dies, Ed. of Philébe xxii; Wilpert, Phileb. 576.
2 Also Empedocles and the Sophists. It is true that the ‘stranger from Elea’’ was in later
times called a Pythagorean (Clem. Al. Strom. 0.48.2).
speculation, or the tradition of the Old Academy is, froma historian’s
point of view, fiction!”
The decision can scarcely be in doubt: Speusippus, Xenocrates, and
Heraclides equate the doctrine of their master Plato, and therewith also
their own philosophical positions, with the wisdom of Pythagoras,
whereas Aristotle sees both Academic and Pythagorcan philosophy
from a certain distance, and even from the stance proper to an active
polemic.!! Plato’s pupils ignore the intellectual developments of a
century and a half, a period of tremendous intellectual turmoil; their
picture of Pythagoras is, in the history of thought, an impossibility.
The consequences of this decision must be faced, without further
attempts at compromise. The “‘derivation system” is an achievement
of Plato and the Academy, a genuine transposition platonicienne’”? of
an older, Pythagorean number philosophy, making use of some
improvements introduced in the circle of Archytas. The tradition that
attributes the “derivation system” to Pythagoreans, or even to Pythag-
oras himself, is to be taken for what it is: evidence for the Old
Academy, for Speusippus, Xenocrates, and Heraclides, but not for
historical Pythagoreanism.178 This involves, as mentioned above,
nearly the whole of the post-Aristotelian tradition on Pythagoreanism.
At least in the philosophical realm,!”4 the only usable evidence for
“0 The contradiction of the two traditions has mostly been overlooked, except by
Zeller; it was only the interest devoted to the Platonic number philosophy that brought
it clearly into the foreground. To be sure, people have often perceived the connection
between the later tradition and Platonism. (On Sext. Emp., above, n. 4; but Raven, CQ
1951, 147ff, saw in Sext. Emp. Math. 10.248ff at least partially Pythagorean material,
perhaps going back to Archytas. On Aétius, cf. Frank 260 n. 1.) Less clearly seen was the
contradiction with Aristotle. Delatte (Vie 234) asserts that the Hypomnemata show aucune
influence of Platonic doctrine, in spite of the Indefinite Dyad in section 25; on the other
hand Wellmann (Hermes 1919, 227) thought he could prove by Aét. 1.3.8 that the Indefi-
nite Dyad was genuine Pythagoreanism. The two testimonies stand or fall together. |
(Rightly, on Hypomn. 25: Wiersma, Mnemosyne 1942, 97; Festugigre REG 1945, 10ff,
with refs.) Often, too, the difference is simply ignored, as by Cornford, PlParm 3ff.
“1 Similarly, the Platonists always present Parmenides as simply Platonic, while
Aristotle emphasizes the Eleatics’ “‘primitive”’ inability to get beyond sense perception
(αἴσθησις) (DK 28A24-25).
172 A phrase coined by A. Diés, Autour de Platon (Paris, 1927) 40off.
“8 Throughout antiquity, and beyond, the interpretation of Pythagorean doctrines
offered by Plato and his disciples held the field; but, though the stream that was later
accepted as Pythagoreanism came from Plato, one ought not to designate Plato and his
pupils as Pythagoreans. They are thinkers sui generis, and later ‘Pythagoreanism”’ is
very largely a dimidiatus Plato. This book comprehends as Pythagoreanism that which was
not “transposed” by Plato, that which existed before Plato and contemporary with him as
the intellectual position and the way of life of persons who called themselves Pythagoreans.
(On the origins of ‘‘neo-Pythagoreanism” see below, ch. I 4.)
174 We shall see in chapters IV and VI that the character of the tradition is no different
in the fields of mathematics and astronomy.
82
what is Pythagorean, astde from pre-Platonic testimonies and Aristotle,
is in statements that agree with them and are not contaminated with
Platonic material.'®
One might still hesitate to accept a conclusion that leads to disagree-
ment with the overwhelming majority of the sources. But it is sup-
ported by Plato himself; the significance of this evidence becomes
clear only when it is seen as foreshadowing the conflict of interpretation
which arose among Plato’s successors.
4. PYTHAGOREANISM IN PLATO AND THE ORIGIN IN PLATONISM
OF THE PYTHAGOREAN TRADITION
Modern scholarship on Plato has taught us to see, in the totality
of his work and in the individual dialogues, the inner consistency and
the necessary unity. The tendency to look for concrete historical
relationships, for the origin of particular inspirations, or the target
of specific allusions, has receded into the background. Not the least
important cause of this is a certain resignation, for it is quite impossible
to determine and delimit, from the study of Plato alone, all of his
“sources.” In Plato, every thesis or argument derives its importance
from its truth value, not from its origin in one source or another.
Foreign material is no longer foreign, but an integral part of the
Platonic structure. This is why the question of the nature of historical
Pythagoreanism is perhaps hardest of all to answer, from Plato alone.
Plato’s dialogues do not suggest strongly that Pythagoreanism
was the determinative influence upon him. Aside from Socrates, three
thinkers in particular stand out: Heraclitus, Anaxagoras, and Par-
menides.? The importance of Parmenides seems to increase in the later
dialogues: alongside the one Timaeus stand Parmenides, Sophist, and
Politicus. All the same, the few references to Pythagoreanism are of
special relevance in their Platonic context. |
Pythagoras is named in a single passage in the Republic, in the
final reckoning with Homer. Has Homer, Socrates asks, earned the
laurels of a lawgiver like Lycurgus, Charondas, or Solon, or even
175 This result is decisive in the case of the Philolaus fragments (ch. III, below). To be
sure, bits of genuine Pythagoreanism make their appearance in the Platonists’ tradition,
as with the tetractys (above, n. 120). But in each case authenticity must be proven.
1 Cf. Diés, Ed. of Philébe xxii; Wilpert, Phileb. 576. Ν
2 Also Empedocles and the Sophists. It is true that the “stranger from Elea” was in later
times called a Pythagorean (Clem. Al. Strong. 1.48.2).
83
those of ἃ practical adviser like ‘Thales or Anacharsis? Or (600 a-b),
if not in public life, is Homer said in private life to have been, during
his lifetime, influential in the education of any persons who cherished
him and his association with them and passed on to their successors
some kind of “Homeric way of life,” as did Pythagoras? He was
greatly loved, in this way, and his followers even to the present day
speak of a “Pythagorean way of life,” and seem in some way to
stand out from the rest of mankind.
What gives this passage its importance is its connection with Plato’s
own lot. For him, too, the most desirable career would have been great
influence in the polis; after this was denied him, he decided to be,
at least “in private life . . . influential in the education”’ of individuals:
he founded the Academy.?
In contrast to this “ private circle” of Plato, the character Timaeus
in Plato’s dialogue is shown as one who combines political activity
and experience with profound philosophical learning, so that he
exemplifies the synthesis of philosopher and statesman which was
Plato’s highest ideal. To be sure, he is accompanied by Critias and
Hermocrates, and it is possible for the hypercritical to point out that
he is not characterized expressis verbis as a Pythagorean.® But for a man
from Italy who has been deeply engaged in studying “the nature of
the universe’ (περὶ φύσεως τοῦ παντός, 27a), it ishard to think of any
other identification than with the ᾿Ιταλικοί, καλούμενοι Πυθαγόρειοι. It
can also be inferred from his praise for the lot of the συνήκοοι of a
great man in the Laws (711e) that Plato was inclined to see his ideal of
the philosopher-king as realized in the circle of the Pythagoreans; for
this ἅπαξ λεγόμενον is obviously the Attic version of the Pythagorean
ὁμακόοι (cf. below, ch. II 4). There is no reason to regard Timaeus as a
historical person. Frank speculated that he served as a mask for Archy-
tas, but the fact that his home city is given as Locri Epizephyrii calls
Philistion to mind.’ It has already been mentioned that even Speusippus
3 The relationship between the Academy and the Pythagorean society has often been
emphasized. Boyancé brought out the significance of their common cult of the Muses
(Muses 249ff). Cf. also Morrison, CQ 1958, 211f.
4 20c-d.
5 A. Rivaud, Timée (Budé ed., Paris, 1925) 18.
6 Above, ch. 1 2, n. 6; cf. Pl. Ep. 7.338c; above, ch. I 1.
’ Frank 128f, with notes 375, 375a, 376. There is nothing among the testimonia collected
in the Timaeus chapter of DK (49) which might not have been derived from Plato. The
argument sometimes heard, that the characters of Plato’s dialogues, and especially the
title characters, are regularly historical, is in all probability not valid for the Philebus, and
therefore not generally. (Cf. also Wilamowitz, Platon II 84, Cornford, Tim. 2f.)
84
and Xenocrates thought of the doctrine of the ‘Tinacus as Pythagorean
wisdom.® All the same, for historical purposes the doctrine of the
Timacus can serve as a source for Pythagorean doctrine no more than—
or rather, just as little as—the Parmenides can for the historical Par-
menides;® and the theory of the elements can be shown to be non-
Pythagorean.!° The fact, though, that Plato uses an Italiote as the
spokesman for his own cosmological scheme, is an indication that in
Magna Graecia Plato had at least found an impulsion or an inspiration
toward a view of the cosmos that seemed to him significantly different
from the system of Anaxagoras.
As with “Pythagoras,” Plato names “Pythagoreans’’ in just one
passage. They call music and astronomy ‘sister sciences,” he says, and
try to find the numbers that represent musical intervals? What Aristotle
says about the “harmony of the spheres’’ is here given additional
depth; rather than criticizing absurdities, Plato sets forth the leading
idea. We are also taken beyond Aristotle by the indication that the
theory of the numerical nature of the intervals was peculiar to the
Pythagoreans and distinguishes them from other, non-Pythagorean
musical theorists.
Having come so far, we can confidently point out other Platonic
allusions to Pythagoreans, like a passage in the Cratylus whose im-
portance has been emphasized by Boyancé.’? More important, however,
is a section of the Philebus.
Socrates-Plato attacks the problem of ἡδονή by relating it to the
more general problem of the “one and many,” unity, plurality, and
their mutual interpenetration, not only in the objects of experience
but in the realm of ideas itself (14d et seq.). An ancient tradition, he
says, shows us the way toward solution of the problem:
There is a gift of the gods—so at least it seems evident to me—which
they let fall from their abode; and it was through Prometheus, or
8 Ch. I 3.
® An expression like that of Pl. Parm. 162b, τῷ τε ὄντι τοῦ μὴ εἶναι καὶ τῷ μὴ ὄντι τοῦ
εἶναι μέτεστι, is diametrically opposed to the doctrine of the historical Parmenides.
10 Above, ch. I 3, nn. 115-116. At Tim. 48b (cf. 3c), the derivation of the elements is
introduced as something completely new, which has never previously been discussed.
18 Rep. 530d and $31b-—c; below, ch. V.
12 Crat. 405c; Boyancé, Songe 97 n. 4, Muses 101, REG 1941, 147ff. (Though the
reconstruction, from the Cratylus, of a Pythagorean ‘doctrine d’Euthyphron” goes too
far. Here again the question rises whether the “‘Pythagoreanism” so disclosed, which is
supposed to underlie nearly all the etymologies of the Cratylus, is not rather Platonism.
Plato is playing, in a “ Pythagorizing” vein, with various preexisting forms.)
85
Il PLATONIC AND) PY'THAGOREAN NUMBER THEORY
one like him, that it reached mankind, together with a fire exceeding
bright. The men of old, who were better than ourselves and dwelt
nearer the gods, passed on this gift in the form of a saying: all things
(so it ran) that are ever said to be consist of a one and a many, and
have in their nature a conjunction of Limit and Unlimitedness . . .
(16c; tr. Hackforth),
Our task is, he says, not to proceed immediately from the One to the
Many but to comprchend, stage by stage, the numerical structure that
lies between the One and the Unlimited—and herein lies the difference
between dialectics and cristic—just as the grammarian knows the
number and nature of the sounds which, in the unitary realm of
language, determine the multiplicity of linguistic expression, or as the
musician becomes master of the infinite realm of tones by his knowledge
of the limited number of the intervals. After the special problem of the
Philebus is then once more formulated, the ontology based on the
opposition of πέρας and ἀπειρία is used profitably in its solution.®
In antiquity Syrianus, Proclus, and Damascius saw a connection
between the Limit-Unlimited pair of this passage and Pythagoreanism,
and specifically with the fragments attributed to Philolaus.14 In general,
modern scholars have assumed an almost self-evident Pythagorean
origin for this material, although Frank}¢ defended the view that this
was not a borrowing at all, but a purely Platonic argument, provided,
in Plato’s way, with a poetic-mythical garb. Porphyry, in fact, ex-
plained the Philebus by reference to Aristotle’s On the Good.”
Now, it can be shown from the reports of Aristotle himself that
the doctrines of the Pythagoreans on Limit and Unlimited cannot have
been developed out of Platonism, as Frank thought.!® But aside from
that, we can find a much stronger argument in the Philebus itself. In
the exposition of the example from music, “the men of old,” of πρόσθεν,
are mentioned again. They recognized the number and kind of the
1. Cf. Wilpert, Phileb.; R. Hackforth, Plato’s Examination of Pleasure: A Translation of
the Philebus, with Introd. and Comm. (Cambridge, 1958); Taylor, Phileb. There is a close
relation between Philebus 14d et seq. and Parm. 158b ct seq.; cf. Cornford, PlParm 213.
“ Syrian. Met. 9.37ff, Procl. In Tim. 184.4, 176.29, 11 168.29, Theol. Pl. 1.5, 3.7, Damasce.
Princ, 1 101.3 R.
16 Zeller 1.457 n. 1, 480.1; Ueberweg—Pracchter 306; Taylor, Phileb. 52ff; Dies, Philebe
xxi (with hesitation); Raven, Py 180ff; Hackforth, Phileb. χοῦ,
16 304. Similarly, Bollinger 74. Mondolfo’s citation of individual occurrences of the
words πέρας and ἄπειρον in pre-Socratic philosophy (Zeller Mondolfo 378f) does not
constitute an adequate refutation of this. Ie would be conceivable that Plato himself
followed hints from a number of different predecessors.
7 Simpl. Phys. 4$3.30fF (Arist. fr. 28),
® Above, ch. D2.
4. Pythagoreanism in Plato
intervals which “we have learnt, conformably to the teaching of the
men of old days who discerned them, to call ‘scales’ (éppovias)”” (17d).
They discovered in the bodily movements of the performers a similar
organization, “that must, we are told, be numerically determined and
be called ‘figures’ and ‘measures,’”’ and advanced the view that “this is
always the right way to deal with the one-and-many problem.” Thus
these musical theorists reach out beyond the boundaries of their
specialty,!® adopting a postulate that takes in everything, and the latter
is identical with that of the “men of old,” previously formulated
(16c-d). Now, in the Republic,?° Plato distinguishes two tendencies in
musical theory, and decides that only one of them, that of the Pythag-
oreans, is worth serious attention, because they measure musical
intervals by numerical proportion. In the Philebus, Plato is at first
less precise: τὰ διαστήματα ὁπόσα ἐστὶ τὸν ἀριθμὸν τῆς φωνῆς ... καὶ
ὁποῖα (17c-d). This looks like mere classification, such as one also
finds in non-Pythagorean musicology; but the following expression,
καὶ τοὺς ὅρους τῶν διαστημάτων is a technical term in the theory of
proportion.2! Rhythms and meters are measured “‘by numbers”
(17d); and coming back to this theme later (2sd-c), Plato states un-
mistakably that musical harmony depends on numerical proportions.
Thus it is natural to suppose that from the beginning Plato was
thinking of the same musical theorists;?* from what Plato himself says
it emerges that the ontology of the Philebus has its roots in Pythagorean-
ism.
So we come into possession of a piece of pre-Aristotelian evidence
for a Pythagorean philosophy of some scope, musically oriented. A
prime necessity, however, is to distinguish precisely between what
Plato inherited and what it became in the alembic of his own mind.
But a complete answer to this problem cannot be derived from Plato's
words alone. The best we can achieve is a lower limit, so to speak:
what is derivative is that which does not follow from the context and
the purpose of the dialogue. Insofar as what we learn in this way
19 There is no indication of any such thing in the treatment of “grammar” (18b-d).
20 s30d et seq.; above, n. 11.
21 In the system of Aristoxenus, too, one finds ὅροι; they are the notes, with their names
(Aristox. Harm. 49.20 M., Pl. Rep. 443d). But these are only ‘‘comprehensible”’ in the
σύστημα; the same interval may be exemplified by quite different notes, while in Plato
they are “comprehended” (ἐπειδὰν λάβῃς, 170) before the σύστημα. In the ratio theory
the ὅροι are determined by the interval peculiar to each case (c.g. 9: 8 is a whole tone).
22 Richter (ΒΟΉ) believes that at Phib. 17¢ Plato is rehabilitating the non-Pythagorean
theory of music, and only at 23¢ goes into the ratio theory, from a new point of view.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
coincides with the testimony of Aristotle, the reconstruction may be
regarded as confirmed.
The fundamental problem, that all being is at the same time one
and many, had been formulated long ago (r4c). It is also discussed
in other dialogues and brought into connection with the problems
raised by the Eleatics.?3 Empirical things (γιγνόμενα) had already been
designated, in passing, as unlimited (ἄπειρα, 15b), in contrast to the
“unities’’ represented by the ideas. What is new in the pronounce-
ments credited to “the ancients” (16c) is, first of all, the antithesis of
πέρας and ἀπειρία. This feature alonc is also found at the beginning of
the later passage resuming the argument: τὸν θεὸν ἐλέγομέν που τὸ μὲν
ἄπειρον δεῖξαι τῶν ὄντων, TO δὲ πέρας (23). In the conclusion drawn from
this (16d), the opposition of one and many again comes to the fore:
the method put forward as ancient tradition is that of Platonic diaeresis,
which Plato was already calling for in the Phaedrus. Still, the special
emphasis on “number” as intermediary between the “One” and the
“Infinite’’ is new. In the example from music theory this “number”
suddenly appears with a different function: the measures and tones are
not only “‘diaerctically” divided, but “measured” in numerical
proportions. When the argument is resumed, the “mixture” of Limit
and Unlimited consists in the fact that “number” is introduced into
things: numerical relations like equal, double, etc. (25d). It is not only
musical harmony that depends on this process, but health of body
and the proper rhythm of the seasons—thoughts already brought
forward by Eryximachus in the Symposium.™* Still different is the
function of “number” in another context of thought: everything
dependent on τέχνη was discovered by its means (16c). Here it is a
question of counting, measuring, weighing; one is reminded of an
allusion in the Politicus to the thesis of πολλοὶ τῶν κομψῶν ... ὡς dpa
μετρητικὴ περὶ πάντ᾽ ἐστὶ τὰ γιγνόμενα.25 So right from the be ginning there
23 Esp. Part. passim, Soph. 2518
*4°186a (medicine), 187a (music), 188a (seasons). There it is Eros that is named, not
number; the speech is not simply Pythagorean, but Platonic artistry in exposition. But
Pythagorean themes do seem to be used.
26 Pol, 284c¢. Plato dismisses these people lightly and ironically, because they are not
used to looking at things κατ᾽ εἴδη. He takes over their leading ideas, but only in a signifi-
cantly refined form, What Plato develops out of Pythagorean thought in the Philebus is
what he misses in the Pofiticus, All the same, it may be that Pythagoreans are in his mind
here too (as scholars have mostly supposed, since Campbell; ch Raven, Pyll 186f), In
one case the fruittul development of the basic idea takes his attention, in the other a more
casypoing, popularizing. application of it is rejected (μι, in medicine; see below, ch.
Ul 4).
4. Pythagoreanism in Plato
is interpenetration of various factors. What is Pythagorean, because it is,
in the Platonic milicu, novel and a bit awkward, is the pair πέρας-ἀπειρία
as well as a certain emphasis on number and proportion. More we
cannot say, from the analysis of Plato’s text; but if we add the testimony
of Aristotle, his words and Plato’s complement one another. The
conception of the world as a harmony of Limit and Unlimited, per-
meated by number, that great creator of order, was for Plato a point
of departure and a guidepost on his quest for intellectual mastery of
the riddle of the plurality and unity of being.”
When the argument is resumed (230), ἃ god” is again named as
revealer of the pair of opposites ἀπειρον-πέρας. Plato, independently
(τιθώμεθα), posits a “mixed” class and also introduces, as a novelty,
a fourth class (the airéa). Here, then, we are on Platonic ground.
This is where the Unlimited first receives its more precise specification,
which could not have been foreseen. Whereas it was previously matched
with plurality (πλῆθος, 16d, 17e), it is now explained with the help of
pairs of opposites like warmer and colder, strongly and mildly, more
and less, rather and less rather (24a). There is emphasis on both the
importance and the difficulty of this way of thinking: “‘if things are
said again and yet again, there is some prospect of the two parties to a
discussion being brought to a tolerable agreement” (246). In every
“more and less rather” lies “‘the nature of the Unlimited” (24e). The
place of a simple conception of a spatial or numerical “Unlimited” is
taken by the continuous ““indefiniteness” on both sides of the limiting
measure. The place of the Unlimited is taken by the Indefinite Dyad,
as we may express it in the light of the reports about On the Good.??
The analysis of Being in the Philebus is a foreshadowing, an early
beginning which makes use of the thought of others, moving toward
the ontology developed in On the Good.?* As Hermodorus expresses it,
ae a ¢ 4 3 3 } ,
τὴν ὕλην ὁ Πλάτων κατὰ τὸ ἄπειρον καὶ ἀόριστον ὑποτιθέμενος ἀπ᾽ ἐκείνων
26 Kucharski, “Le Philébe et les Eléments harmoniques d’Aristoxéne,” Εόν. philos. 84
(1959) 41-72, and H. Koller, “Die diharetische Methode,” Glotta 39 (1960) 6-24, believe
that Plato is developing the method of diaeresis according to the model of music theory,
but a contrary argument would be that there is nothing in the Phaedrus, Sophist, or
Politicus to suggest any such connection (though in Phdr. 270d he speaks of "5 numbering
the forms), whereas the feature which is peculiar to the Philebus, πέρας -ἀπειρία is attested
by Aristotle as Pythagorean. . .
2? Hermodorus’ evidence is at ἢ. 29 below. Merlan has drawn attention, rightly, to the
role of the problem of τὸ μᾶλλον καὶ ἧττον δέχεσθαι in Aristotle’s doctrine of the cate-
gories (Philologus 1934, 35ff). Even modern writers are unanimous that there is a relation-
ship to the Ou the Good; cf. Taylor, Phileb. soff; Ross, Met. 1 171; Wilpert, Phileb.
passim. ce |
# On questions remaining open in the Philebus, cf. Wilpert, Phileb, §80f.
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
αὐτὴν ἐδήλου τῶν τὸ μᾶλλον Kai τὸ ἧττον ἐπιδεχομένων, ὧν καὶ τὸ μέγα
καὶ τὸ μικρόν ἐστιν.39
What Plato presupposes as “source” is no more, and nothing else,
than what Aristotle treats as the doctrine of the Pythagoreans. There is
nothing of the One and the Indefinite Dyad, but Limit and Unlimited;
and if we are right in combining with the Philebus the passage in the
Politicus, no theory of ideas. At the same time, we sce Plato’s philosophy,
emerging from these Pythagorean stimuli, moving in the direction of
the ontology of the lecture On the Good. Our decision, regarding the
tradition about Pythagoreanism, between that of the Old Academy
and that of Aristotle, is thus irrevocably confirmed: it was not Speusip-
pus, Xenocrates, and Heraclides, but Aristotle, who gave authentic
information on the Pythagoreanism that was there before Plato. And
what is more, we can understand, from Plato, how the former tradition
arose, with its distortion of the historical picture.
For Plato’s affirmation of the divine origin of the doctrine of Limit
and Unlimited is more than a glittering sequin on the fabric of the
exposition. It signifies that its truth is beyond doubt; and Plato feels
that this imposes on him the obligation to grasp the truth of this idea
and its all-encompassing significance. Such a divine revelation is not
something finished and complete, but a task to fulfill—like that of
“coming to the aid of the god” (βοηθεῖν τῷ θεῷ) in the Apology.?° It is
precisely because of the authority of such a doctrine that interpretation
must sct in immediately; and in this process Plato’s own thoughts and
those of others become almost inextricably intertwined.
Herc is a type of interpretation untroubled by minutiae of historical
accuracy and only interested in the sense intended. Actually, the Greek
question τί λέγει; does not mean, “‘ What words are used?”’ but “ What
does this mean?” This method is characteristic of Plato, but surely not
of him alone.3! When, in the dialogue named after him, Protagoras
speaks of Homer and Hesiod, Orpheus and Musacus as Sophists,
Socrates overtrumps him with the claim that the most σοφισταί,
students of φιλοσοφία, are to be found in Crete and Sparta.2 A similar
passage in the Laws is more scrious. At first the Spartan and Cretan
constitution is represented as aimed toward warfare as its only goal
29 Simpl. Phys. 247.30ff.
Ap. 21b,
“Cf. the sharply ironical, but by no means entirely inaccurate, formulation of Cherniss,
referring to Aristotle (Riddle 30): “Aristotle is one of those who cannot be refuted by an
author's words because be is sure that the wuthor was unable τὸ say what he really
thought.”
Prot, g1r6d, pq2ab,
4. Pythagoreanism in Plato
(625d). This object is rejected as comprehending only one portion of
ἀρετή. The result of this, however, is not rejection of the Cretan and
Spartan vézor—thcy are of divine origin, handed down by Zeus and
Apollo. Therefore, ὥσπερ τό τε ἀληθὲς οἶμαι καὶ τὸ δίκαιον. ὑπέρ γε
θείας (πολιτείας) διαλεγομένους λέγειν (630d-e), the interpretation must
be revised; and to the bewilderment of the Cretan and the Spartan, the
object of their constitutions appears all of a sudden as much more
comprehensive, and by no means restricted to the military aim.*
This is the spirit in which, on the basis of the Philebus, we must
judge the relationship with Pythagoreanism; here too the tradition
is set before the philosopher as a task which has to be thought through.
And as the ancient claim, that Sparta’s constitution came from the god
of Delphi, must be taken seriously, the “ divine” origin of Pythagorean
tcachings, too, is more than a facon de parler. Aristoxenus tells us that
Pythagoras got his doctrines from the Delphic oracle,** but Aristotle
says that Pythagoras himself was believed to be the “Hyperborean
Apollo.”> Taken together with the Philebus, this can only mean that
the Pythagoreans, even as early as those whom Plato knew, understood
their own philosophical activities as developments of the basically
identical doctrines of their master, the divine Pythagoras, “but that it
all comes from that great man (εἶναι δὲ πάντα ἐκείνου τοῦ dvdpds).’ 8
We must suppose, then, that this way of thinking was already estab-
lished in Plato’s day. It explains the surprising uncertainty of Aristotle
as to the chronological relationships of the Pythagoreans.®’ He refused
to accept the assertion that all this was Pythagoras’ teaching, but he
had no basis for more precise chronological determination (and perhaps
no interest in it).
But where Aristotle looks upon it with critical aloofness, Plato's
disciples join him in taking their place within the Pythagorean tradition.
In this respect too, the Philebus confirms the conclusion we were able
to reach from the contradiction of Aristotelian and Academic accounts
of Pythagorcanism. Plato’s school sees in its own philosophical treat-
ment of the problem of ultimate principles a continuation of Pythag-
oreanism, so that ancient material is reinterpreted accordingly.
This Platonic interpretation of Pythagorean philosophy became
MOL also W. Keber, Platos Stellung zur Sparta~Ideologie (Diss. Miinster, 1057).
M4 Aristox. fr. 1g DVL. 8.8., 21.
"Δ Below, ch. ΠῚ 3. oe
tam. PP RR Conun. math. sc. 77.225 ef below, ch. ID 5. lam. VP 198: καλὸν δὲ τὰ
4 ‘ td at OF γὼ"
mivra Πυθαγόρᾳ ἀνατιθέναι τε καὶ ἀπονέμειν καὶ μηδεμίαν περιποιεῖσθαι δόξαν ἰδίαν...
(similarly ibid. 158). The proverbial αὐτὸς ἔφα D.L. 8.46 (with the parallels given Delatte,
lie ad ἴσο),
? Above, ch. 12, nt gy, 120,
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
dominant in the ancient tradition. Aristotle was the only one to con-
tradict it, and shows us thereby what had been there before Plato;
and in fact what Plato presupposes is what Aristotle criticizes.
It is not necessary at this point to pursue in detail the further allusions
to Pythagoreans in Plato, whether certain or conjectural.38 What has
been said is enough to serve at Icast as a sketch of the general trends.
The true problem of the Pythagorean tradition lies in Platonism, for
Platonizing interpretation took the place of historical reality. One
can only guess at the reasons why Plato and his pupils saw themselves
as continuators of Pythagoreanism. Personal contacts were certainly
important; they were already present in the Socratic circle. Simmias
and Cebes of Thebes, “hearers” of Philolaus, appear even in Xenophon
as pupils of Socrates,?® and we may believe that, as the Phaedo has it,
the Pythagorean Echecrates of Phlius had some connection with
Socrates.*° Plato’s friendship with Archytas is attested by the seventh
Letter,*! and the later biographies of Plato agree that a principal motive
of his first journey to Magna Graecia was to establish contact with the
Pythagoreans there.*? One must proceed cautiously in trying to deter-
mine what extent of doctrinal agreement there may have been between
Plato and the Pythagoreans. There is a considerable danger of projecting
onto the Pythagoreans, unjustifiably, the unique style of thought and
presentation that were Plato’s. We must remember the possibility of
individual influences that could be at work in the impressiveness and
magnetism of such a personality as Archytas. It is worth remembering
that in the seventh Letter immortality figures as a belief not subject to
skeptical doubt (33 sab; cf. Rep. 498d). In addition to external influences
38 On Philolaus, below, ch. III 2; on the ψυχή-ἁρμονία doctrine, below, ch. ΠῚ 2; on
the harmony of the spheres, below, ch. IV 4; on the “nuptial number,” below, ch. VI 4;
on the origin of the soul, ch. V; on Gorg. 493a, below, ch. HI 2; on Gorg. 507e, above,
ch. I 3, nn. 155-157.
39 Pl. Phd. 61d-e, Xen. Mem. 1.2.48, 3.11.17.
10 Echecrates as a Pythagorean: Aristox. fr. 18-19 (cf. DK 53); ps.-Plato, Ep. 9, where
Echecrates appears as a friend of Archytas, is doubtless a rhetorical exercise with no
historical authority.—The Echecrates from whom Timaeus claims to have received
information about Locrians (FGrHist 566F12=Polyb. 12.10.7) is, in spite of the chrono-
logical problems, likely to be the same as the Pythagorean from Phiius (cf. Oldfather,
RE Supp. Hl 417£; Jacoby Ib, Kom. $52, Noten 326, 195; Echecrates of Locri as a teacher
of Phito: Cic. Fin. §.87, Val. Max. 8.7.3). Cf. also below, ch. H 5.
41 338c, 330b.... ᾿Πρχέδημον, ὃν ἡγεῖτό με τῶν ἐν Σικελίᾳ περὶ πλείστου ποιεῖσθαι, τῶν
"Apxtryn συγγεγονότων ἕνα. According to this Archytas had “pupils.” Cf. alse. 3500.
1.1L. 3.6, Cic. Mine 5.87 (from Antiochus), Rep. ἀντ, Apul. Plat, 1.3, etc. (with
inconsistencies in details). Cl also Wilamowitz, Plafon ΠῚ ΒΔ In Anon. Phot. 4188 18
Plato is called "Apytrou τοῦ πρεσβυτέρου μαθητής, this is polemic against the other view,
that Archytas had been a pupil of Plato (lmaniseh 46). ‘Uhe meaningless senior Archytas
(Apal. Plat, 1.4) and "Apytray ὁ πρεσβύτερος (laut, PP tog) is a imisaunderstanding of the
RAINE SOULCE,
4. Pythagoreanism in Plato
there is Plato’s peculiar hesitation ever to speak dogmatically, in his
own name. When Socrates receded into the background, in comparison
with Plato’s deepening interest in mathematical science, he was glad
to present, as the spokesman of his “likely account” in the Timaeus,
a man from Magna Graecia, even though the cosmic design presented
there went far beyond the words of the predecessors who had helped
stimulate him. Finally, it seems to have struck Plato sometime that a
way toward the solution of the growing problem, in the theory of
ideas, of the κοινωνία τῶν εἰδῶν, opened itself up in the form of number
concepts. And it was in Pythagorean number speculation that he found
this solution foreshadowed.
It is scarcely surprising that Plato’s disciples subsequently saw
l ythagoras only through the eyes of Plato.** But it is also comprehensi-
ble that the doxography took its information about Pythagoras from
the writings of Plato’s pupils,“* when we realize that there was nowhere
clxe a tangible account of the philosophy of Pythagoras, clearly out-
lined and provided with a fixed technical terminology—as was done
tor the doctrine of Parmenides in his poem. There was no “Word of
Pythagoras,” and in the absence of authentic sources whoever wanted
to mention Pythagoras could do nothing but rely on those expositions
that spoke confidently about a “philosophy of Pythagoras.” Aristotle
spoke only of “Pythagoreans,” aside from the fact that the material
was scattered through Aristotle’s works and presented in polemical
fashion, while the positive accounts of the Platonists provided con-
veniently organized reports. In addition, the Platonists’ expositions,
ax the only ones to suit later philosophical taste, were for that reason
nore likely to give an impression of authenticity.”
liven in the Life of Pythagoras which Photius excerpted,® Pythag-
areanism, Academy, and Peripatos are seen as making an unbroken
uty: the ninth successor of Pythagoras is Plato, the tenth Aristotle.
lot the most part, though, Plato and Pythagoreans are separate in the
later tradition. The Pythagorean tradition freed itself, cut loose from
the Academy, and as a result of this separation from its origin it was
M Speusippus and. Xenocrates also had personal contact with Pythagoreans; above,
rhe Poa, an 10}.
onthe problem of the sources of the Vetusta placita, see ch. 1 2, n. 76, ch. I 3, nn. §6,
Hit, a4?
bor the playing off of “imore genuine”? Pythagoreans against Aristotle, sce below,
eh Ut. With reference to the reports of Aristotle, Philoponus clearly formulates his
principle for interpretation of Pythagoreanism (De ant. 70.2): εἰ μὲν yap τὸ φαινόμενον
ἐκ ληψόμεθα, καταγέλαστον ἔσται καὶ οὐ συφῶν, ἀλλὰ γραῶν λῆρος" εἰ δὲ ἐκεῖνοι σοφοί, δεῖ
ἔπλει" re τεὴν φαινομένων σεμνότερον,
“4 100 1 (ch above, eh. 14. ν.)
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
able to develop a life of its own. One cause for this lies in the fact that
tradition about Plato’s oral teaching lost in importance, in comparison
with the steady influence of the Platonic dialogues. The doctrines
formulated in the latter were well known, while the theory of ideal
numbers sank into desuetudce.
The most important factor, however, was the effect of the direction
of Academic development on the tradition about Pythagoras. From
Aristotle's time it had been customary to see Plato as a synthesis of
Socrates and Pythagoras.*” In the generation of Plato’s pupils the
“Pythagoreanism,” metaphysical speculation, carried the day. But,
when a reaction set in and the Academy became predominantly
“Socratic,” Pythagoras necessarily lost his place. This is just what
happened, from the time of Arcesilaus and the ‘‘ Middle Academy ”’;
the school of Plato went over to skepticism, and there had to be a
parting of the ways with Pythagoreanism.
There were necessarily two aspects to this breach. Insofar as the
Academics following the Socratic-skeptical trend still felt themselves
to be Plato’s successors, they had to push diligently aside everything
“dogmatic,” and especially the mathematical-scientific and metaphysi-
cal teaching of the school, as not genuinely Platonic. They found another
origin for it: Pythagoras. On the other hand, what this process dis-
carded retained, even for the rationalistic Hellenistic world, a certain
fascination. Those who were attracted by it could no longer attribute
it to Plato, against the authority of the Middle Academy, but found it
necessary to reach back for the authority of “‘Pythagoras.”’ And when
he took the limelight, Plato and his pupils were stigmatized as plagiarists.
To Cicero, “Academic” and “skeptical” mean the same thing, and
in fact Plato was often invoked by the skeptics.4* This tradition may
have affected the report of Sextus Empiricus on the theory of ideal
numbers : 49 use was made of an exact transcript of the lecture On the
Good, but the whole is presented as refutation of the Pythagoreans.
Plato’s name first occurs only quite incidentally (10.258), but in the
skeptical refutation he is cited at length (10.302ff). Criticism of the
theory of ideal numbers takes the form of a struggle, in alliance with
Plato, against Pythagoras.
“ Arist. Met. 987a2otf, cf. Dicacarchus fr. 41, Cic. Rep. v.16, De or. 1.42, Fin, §.87,
Fuse. §.10, Numentus ap. Euseb. Pracp. evang. 14.5.9, Aug. De civ. D. 8.4.
“Cic. Acad. 2.74, 1.46, Sext. Emp, PH 1.221, Proll. in PL τὸ. 4-8 (p. 20§ Hermann).
Ct Burkert, “Cicero als Platoniker und Κορ ον," Gynmasian 72 (1965) 175-200. - ‘The
first ps.-Xenophontic letter (Hercher p. 788) is also a polemic of the “Socratics” μηδ
Plato's" Pythagorizing.” "The reproach is made against the Platonists: Δγύπτου ἠράσθησαν
καὶ τῆς ΠΙ|υϑαγόροιν reparddous σοφίας.
* Above, ch. Pg, παν gif
4. Pythagoreanism in Plato
The contrary position is expressed in Numenius’ work On the
Difference (διαστάσεως) between the Academics and Plato.°° Por Numenius,
Plato and “the great Pythagoras” are about the same thing, and
whatever Socrates had to offer also came from Pythagoras (5.7).
Plato’s direct disciples had followed their master, and “it was primarily
because of them that Pythagoras came to be highly honored” (5.2);
Arcesilaus and his pupils had wandered from this path, and with scorn
and anger Numenius condemns this apostasy.
But long before Numenius this attitude had produced remarkable
results. From the third century B.c. on, the apocryphal Pythagorean
writings appear, each trying to outdo the other,®" presenting Platonic-
Veripatetic doctrines as original pronouncements of Pythagoras and
his pupils. The interpretation of Pythagoreanism that Speusippus,
Xenocrates, and Heraclides had given is hypostasized in this revival,
and the purported originals make the intermediaries superfluous.
‘Thus the criticism of Pythagoreans themselves is turned against the
Platonists:
.., that Plato and Aristotle, Speusippus and Aristoxenus and
Xenocrates, as the Pythagoreans say, appropriated what was fruitful,
with slight modification, but collected some superficial or incon-
sequential things, whatever is brought forward by those later
malicious slanderers in an effort to refute and mock the school, and
put these down as the special doctrines of the sect . . ὃ
‘Thus even these Pythagoreans admit, implicitly, that what was essen-
tial, or “fruitful,” in their doctrines agrees with Plato and Aristotle,
and that they could make no good use of what historical tradition had
tootter asthe ἴδια of Pythagoreanism. Later Pythagoreanism is stamped
Nunmienius fr. 1ff=Euseb. Praep. evang. 14.5-9.
On the apocrypha, see Burkert, Philologus 1961.—Antiochus of Ascalon led the
Academy back to dogmatic ways, and left skepticism, in turn, homeless. een
in just as unbistorical a way, it sought out a new founder to whom it might trace itself,
Vyrtha, (( A. Weische, Cicero und die Neue Akademie [Minster, 1961] rosff)
Way Πλάτωνα καὶ ᾿Αριστοτέλη Σπεύσιππόν τε καὶ ᾿Αριστόξενον καὶ Ξβενοκράτη,
vig φασιν οἱ Πυθαγόρειοι, τὰ μὲν κάρπιμα σφετερίσασθαι διὰ βραχείας ἐπισκευῆς, τὰ δ᾽
ἐπιπόλαια καὶ ἐλαφρὰ καὶ ὅσα πρὸς διασκευὴν καὶ χλευασμὸν τοῦ διδασκαλείου ὑπὸ
rap ἢπυκίνως ὕστερον συκοφαντούντων προβάλλεται, συναγαγεῖν καὶ ὡς τς τῆς
alpdveay καταχωρίσαι... (Por. VP 43). The source of this passage (transition rom
Mauleratas to Nicomachus) is impossible to determine. ἀνασκευὴν Is my suggestion for
MOS ῥιασκεμνήν; cf Sext. Emp. Math. 6.4, 8.196, 9.§§, etc. (P. Shorey, CP 27 [1932] 1758,
proposed! διασυρμόν or διαστροφήν, cach less likely on paleographical grounds.) i ᾿ πο
toe, without qualification, that Plito, Speusippus, and Xenocrates only left the Pythag-
nieate ἐπι πίλαια, Plato makes almost no explicit pronouncements about Pythagorean-
br, while Speasippus and Xenocrates translate and modernize it, This ts noe Phony
however, Co satisfy the “imperialistic” attitude of the Tater Pythaporeans. On Plato's
“appropriating Pythagorean doctrines, see also 6 PP gt,
I. PLATONIC AND PYTHAGOREAN NUMBER THEORY
so deeply with Platonism that it has no longer any conception of its
real origin.
One might therefore define later Pythagoreanism as Platonism with
the Socratic and dialectic element amputated. In fact, Plato remained
the principal source for all later Pythagoreans—Plato’s myths, and in
particular the Timaeus. The apocrypha, presenting the supposed
originals, could make no headway against this overwhelming
influence. Apollonius of Tyana taught “doctrines about the physical
world similar to the opinions of Plato’s Timaeus,” and though Proclus
considered the “Timaeus Locrus” document genuine and put it at
the head of his commentary on the Timaeus, it was not this sorry
scribble but Plato’s Timaeus itself that he classified, along with the
Chaldaean Oracles, as far excelling all other literature.5* Scholars have
shown in different ways that Neoplatonism is quite closely dependent
on the Old Academy, and “‘Pythagoreanism”’ too belongs in this
category. It is also basically Platonism, existing at a time when Plato
(as interpreted in Pythagorean fashion) had lost his position in the
Academic school. Later, neo-Pythagoreanism converges, in the
philosophical realm, with Neoplatonism.*
And yet Pythagoreanism is not sufficiently characterized by that
which is lacking, as compared with Plato. The inherited material that
was lumped together under Pythagoras’ name, undigested or mutilated
though it may have been, was in this process raised to a new dignity
and endowed with unexampled authority. The whole body of apo-
cryphal literature lics within the realm of religion. The ontology of the
Old Academy was oriented toward the divine; the Pythagorean
pscudepigrapha meet a subconscious religious need of the Hellenistic
period. And when, from the first century B.c. on, people once more
come forward to declare themselves Pythagoreans, their most noticeable
characteristic is that they are seeking (or even, as for example in the case of
Apollonius of Tyana, claiming to possess) a superhuman, divine wisdom.
And it may be that in this very point—not in details of doctrine but
in the claim to possess divine knowledge—we are most likely to find
an clement of its real origin, in the influence of Pythagoras of Samos.
δ Philostr. VA 6.22, Marinus Κ΄. Procl. 38.
δ4 Dodds, CQ 1928, Merlan, PlNeop!.
6) Alongside Pythagoreanism as doctrine stands Pythagoreanism as a way of life.
While the Hellenistic apocrypha promoted Pythagoreanism ina literary way, the real
point of “neo-Pythagoreanism” from the times of Nigidius Figulus and Anaxilaus of
Larisa was to take it seriously in. everyday life. Once more there were people who devoted
themsclves to a Bios Πυθαγόρειον (Burkert, Philologus 1961).
II. Pythagoras in the Earliest Tradition
I. SOURCE PROBLEMS
Platonizing interpretation brought basic change to the Pythagoras
tradition, so that a historical reconstruction of what it was like before
Plato’s time must be based on the pre-Platonic evidence. This can be
supplemented by reports that were not drawn into the process of
Platonic transmogrification; but, to judge these with any approach to
confidence, we must first survey the nature of the tradition as a whole.
lt is no longer admissible simply to reject what is late attested, as
Zeller did; any such item might have its origin in an ancient and
authentic source. After Rohde, the most important advances in the
analysis of the sources were made by Delatte and Lévy. Significant
results have been achieved; but the essential thing is to distinguish
between what is certain and what is merely conjecture.’
Most of our material on the life and activities of Pythagoras is
collected in the eighth book of Diogenes Laertius, in Porphyry’s Life
of Pythagoras, and especially in lamblichus. We may add the tenth book
of Diodorus, of which only fragments are preserved, and the very
short sketch in Justin. Photius’ excerpt from an anonymous life of
Pythagoras has very little that pertains to history or biography.
lamblichus is most detailed; his plan was to present the xara Πυθαγόραν
ψιλοσοφία in a work designed to run to ten books, because ten is the
perfect number. The Pythagorean Life (for his subject was the appro-
priate way of living, rather than the biography of Pythagoras as such)?
was followed by the Protrepticus, the book On General Mathematical
Knowledge (De communi mathematica scientia), and, as a treatment of
the special sciences, a commentary on Nicomachus’ Introduction to
! An example of analysis carried too far is Bertermann’s dissertation. In his concluding
table (796), the whole text of Iamblichus’ Vita Pythagorica is traced back, sentence by
sentence, through two or three intermediaries for each, to sources of the fourth century
hc. ut his evidence is mainly vague associations and scarcely ever attains the status of
proof, or an example of the way in which hasty source analysis can lead to error, sce my
discussion of the “letter of Lysis,” Philologus 1961, 17-24.
“Ror convenience, the commonly accepted title Life of Pythagoras has been retained
heie.
Il. PYTHAGORAS IN THE EARLIEST TRADITION
Arithmetic. Later in the series came the book on numerical theology
(Theologumena arithmeticae). Further parts are not preserved, and perhaps
not all of them were written. It was Iamblichus who set the direction
for the later Neoplatonists, toward a definite equation of Platonism
and Pythagoreanism.®
Our first question must be as to the direct sources of Iamblichus.
Here analysis is easiest in the case of the Theologumena arithmeticae;
one source, Anatolius On the First Ten Numbers, has been recovered,
and the second, Nicomachus ’ApiOunrixdy θεολογουμένων βιβλία β΄, we
have in an excerpt by Photius (Bibl. 187). The manuscripts of the
Theologumena arithmeticae often name Anatolius and Nicomachus, and
the book proves to be, essentially, a cento made up from the two older
ones. What is not Anatolius is mostly quotation of Nicomachus,
including his own quotations. The contribution of the author, or
rather compiler—and in spite of doubts this was probably Iamblichus
himself—is merely arrangement and introduction.
Certainty was reached by Erwin Rohde on another point, the rela-
tion between Porphyry and Iamblichus. The latter did not use Por-
phyry directly; instead of the general Philosophic History he used special
works on Pythagoras.> Thus we have access to important sections of
Nicomachus’ biography of Pythagoras, and in their original wording.®
3 It is to be ascribed to the influence of Iamblichus that the chapter on Pythagoras in
Porphyry’s Philosophic History became separated from the rest and is therefore the only
surviving section (cf. Harder xv).
4V. de Falco, “Sui Theologumena arithmeticae,” Riv. indo-greco-italica 6 (1922) fasc. 1/2,
49-61; H. Oppermann, Gromon § (1929) 548-558. If lIamblichus had inserted the source
citations himself they would have been evenly distributed; but, as it is, they are found
almost exclusively in sections based on Nicomachus.
5 That Iamblichus does use Porphyry is maintained, after Zeller 1 365 n. 1, by E. Norden
(Agnostos theos [Berlin, 1913] 344 ἢ. 2) and more recently by J. A. Philip (TAPA go
[1959] 185-194. Rohde’s proof to the contrary (Q 125ff) was based on the fact that the
Nicomachus sections of Porphyry occur, without exception, as verbal quotations in
lamblichus, while apart from them there are only occasional points of contact. In addition,
Iam. VP 233, compared with Por. VP 59, proves that Iamblichus is copying the wording
of Nicomachus, not that of Porphyry. Iam. VP 253 has the Doric forms proper to the
Lysis letter (Hercher, Epistologr. er. p. 603), which are not preserved by Porphyry (58).
lam. VP 170 has Metapontum, correctly, where Por. VP 4 names Croton (Nauck liv, 36;
below, ch. II 2, n. 18). Even Jamblichus could not have spoken of “‘gencral agreement”
(VP 248) if he had had before him the contradictory versions of Por. VP 56-57.
§ Por. 20-31 Tam. 30, 33, (241), 34, 60, 61, 62, 36, 63, 134-135, (142), 136, 64-67;
Por. §9-61t> Tam. 233-237. The material promised in the sentence with which the extant
text of Porphyry breaks off is in lam. VP 189-194. In these cases Porphyry names Nico-
machus: on the other hand, Iamblichus names Nicomachus at VP 251. Purther, Iam. VP
2§2p.13§.10-17Deubner Por. VP ςς p. 47.18-48.t Nauck; fam. p. 139.18-136.13 > Por.
$7f pp. 49.16 so.tr. Thus we can be certain that these passages are all from: Nicomachus
(contra figer, sy, who supposes the source is the “handbook,” as well as Delatte, Pol. 219,
and Minar 68 n. 64, who take Neanthes as the direct source; correctly Rohde, αἰ φῇ,
1. Source Problems
It also becomes obvious that Porphyry copies more mechanically than
lamblichus. Where Porphyry gives two connected excerpts from
Nicomachus, Iamblichus may distribute the same material in different
chapters, according to an artificial outline of his own.” Without the
parallels in Porphyry it would be a hopeless undertaking to attempt to
identify in the mosaic those tiles that make up the contribution of
Nicomachus.
Further, there is in Porphyry a connected passage from Moderatus on
number theory,® and twice he cites Antonius Diogenes’ romance on
The Wonders beyond Thule, though it is not quite clear where the
uiotations end.® His basis for the remainder must be a handbook con-
taining many citations of its sources, and for this part the points of
contact with Diogenes Laertius are especially numerous.
Corssen, Philologus 1912, 332ff, Lévy, Sources 92 n. 1; the fact that Iamblichus puts
passive for active at p. 135.17, and thus has the persecutors being killed, is his own mis-
understanding, (Lévy, Sources 97). Since, further, Iam. VP 248 p. 133.12-14=Por. VP
s4l pp. 46.24~-47.2 show verbal agreement, and in addition Iam. VP p. 13 3.8f introduces
ile version of the Pherecydes episodes which is later (p. 135.11) attributed to Nico-
iMachus, the excerpt from Nicomachus must begin as early as lam. VP 248, and Por. VP $4,
and the Aristoxenus citation (fr. 18) must come from Nicomachus (as fr. 31==Iam. VP
241 Por. VP 59). This had already been deduced by Rohde, from Iam. VP 251: Νικόμαχος
δὲ τὰ μὲν ἄλλα συνομολογεῖ τούτοις (ΟἸ15; Lévy, Sources 116, attributes the Aristoxenus
titation to the third source of Iamblichus, the “‘handbook”’; Diels, DK 14.16 n., names
Apollonius as the source of the Aristoxenus citation, doubtless a mere slip, passed on by
Vunpanaro Cardini, p. 55). Nicomachus cited Aristoxenus, but preferred to follow
another version. Most authorities suspect this is Neanthes, who is cited by name at Por. VP
44 p. 48.1 (von Fritz, Pol. 5, after Corssen), but Lévy thinks it was Satyrus (Sources 61ff;
4, 1).L. 8.40), and that Neanthes is the basis rather for D.L. 8.39=Hesychius. It is not
likely that the citation of Dicaearchus, Por. VP «68 (fr. 34 W.) was in the text of Nico-
μεμα αν (Corssen, Philologus 1912, 341f, according to von Fritz, Pol. 7, already in Neanthes),
sittee lamblichus does not seem to know it (above, ἢ. 5). Nor does Dicaearchus fr. 33=
lw, 1’P 48 appear in Iamblichus; Porphyry has inserted into the Nicomachus text a
passipe from the “handbook” source (the same citation at D.L. 8.40=fr. 35b W.).
'Lamblichus’ skill in combining disparate quotations can best be seen from his use of
lato: cf, Merlan, PlNeop! 148ff—Small variations between Porphyry’s and Iamblichus’
wording are frequent. Generally, Iamblichus is fuller; and it may be that in the verbosity
of his exposition he introduces expansion as often as Porphyry abbreviates.
"4h $3; cf. above, ch. 13.
“Por, VP τοῦ and 32ff; Lydus Mens. 4.42 p. 99 W. shows that Porphyry’s source in
IP a4 is still Diogenes Antonius. Rohde first (Q 126) assigned 10-17 and 32-45 to
Autonius, then later (Rom. 272 n. 2), on the ground that in his novel Antonius could not
have made explicit citation of sources, only 10-14, 32-36, and 44. (In τς Dionysophanes
incited, in 41 Aristotle). On the other hand Jager (36 ff, 43-47) and K. Reyhl (Antonios
Hyovenes, Diss. Tubingen, 1969, 20-31) hark back to Rohde’s original thesis, excluding
ἐγ the direct citations. To be sure, the mention of ἄδυτα connects Por. VP 17 and 34.
lteyhl adds Por. VP 46-47, which is quite improbable, because Iam. VP 228 has copied
the same text, but lamblichus did not use Antonius Diogenes—and Por. VP $4-$5,
riphtly comparing φυσιογνωμονήσας §54 and §13, though this is squeezed into a context
from Nicomachus (above, πὶ 6).
Il. PYTHAGORAS IN THE EARLIEST TRADITION
Another direct source, Apollonius of Tyana,!° is once explicitly
cited in Iamblichus (254-264), and once shown to be a source by
parallel passages in Porphyry." Rohde showed convincingly that two
other continuous sections, the speeches of Pythagoras in Croton!? and
his meeting with Phalaris,!° have the same origin.
Rohde went on to propose a mechanical two-source theory,
according to which Iamblichus drew material exclusively from Nico-
machus and Apollonius; he tried to assign each chapter to one of the
two, though allowing that Iamblichus may have worked out some
individual passages on his own.!4 This two-source theory was vigor-
ously attacked by Méautis, and definitely refuted by Lévy. At least
10 The identity of this Apollonius with the wonder-worker of Tyana was contested,
after Wyttenbach, by Méautis (91), but has probability on its side. Apollonius was a
conscious and enthusiastic Pythagorean.
1 Tam. VP 3-8, 11; cf. Por. VP 2. Rohde (Q 128f) and Lévy (Sources 105ff) give the
whole section 3-25 to Apollonius; but it is more probable that Iamblichus’ compilatory
method begins at the very beginning, though perhaps in an especially careful vein. (Cf.
the connection of Iam. VP 19 with Nicom. Th. ar. 53.1ff.)
12 Jam. VP 37-57 is bracketed with the Apollonius passage 254ff by 4 separate con-
cordances: 49 and 262; the temple of the Muses 50 and 264; the Pythaeum (or Pythium)
§0 and 261; θεῖος, $3 and 255.
18 Tam. VP 215-222; cf., for Apollonius before Domitian, Philostr. VA 8.1ff (Rohde,
Q 164ff; Lévy, Sources 109, Lég. soff). Boyancé (REA 1934) tried to trace the section to
the Abaris of Heraclides Ponticus. In refutation, Miss De Vogel points out the Stoic and
late Platonic terminology (Mnemosyne 18 [1965] 388-396, Pythagoras, 304~306). Festugiére
(REG $0 [1937] 474-476) adduces comparisons with hermetic-gnostic material. The rapid
summary at 219 shows that Iamblichus was not composer but compiler.
14 According to Rohde Iamblichus’ own hand can be seen in 103-105, 157-162,
167-186, 198-199, 214, 223-228, 240-241, 244-247. Where doublets occur, Rohde
always gives the one version to Nicomachus, and the other to Apollonius; but we should
bear in mind the possibility of free self-citation by Iamblichus.—Jam. VP 159f=In Nic.
5.27ff is attributed by Rohde to Nicomachus’ biography of Pythagoras (Q 156f); but
are we to suppose that Iamblichus would really cite Nicomachus to comment on Nico-
machus, and in addition that he would have been able to find such a detailed parallel to
what Nicomachus says at Ar. 1.12 The passage is tolerably well integrated into Iam. VP
159f, while it seems a disconnected, interpolated block at In Nic. 5.27ff; Iamblichus is
repeating his own paraphrase.
16 Méautis 87ff, Lévy, Sources 111ff. E.g., Nicomachus (lam. VP 30=Por. VP 20)
names 2,000 auditors of Pythagoras, Apollonius (lam. VP 254, 260) gives 300, but Iam.
VP 29, like D.L. 8.15, 600. (lam VP 29 is marked as an interpolation by Deubner, but,
given the compilatory habits of Iamblichus, incongruitics are always to be expected.)
Iam. VP 35 had already been assigned to a “handbook” by Rohde (Q 131). Further:
lamblichus presents the miracles of Pythagoras in two versions, one according to Nico-
machus and one according to Aristotle. If it can be shown that the versions taken from
Nicomachus are alinost always altered ina distinctive manner, soas to blunt the paradoxes,
it is quite improbable that Nicomachus himself recorded both versions (as Rohde says,
Q 152f); and why should Porphyry always have chosen the non-Aristotelian formulation?
(CE. below, ch. Π| 4.)—The sentence about the 3 books of Pythagoras (lam. VP 100)
cannot come from Nicomachus, who thought Pythagoras left no writings (VP 57; cf.
lam. VP 146, where doubtless Nicomachus is to be understood as one of the ἐλλόγιμοι καὶ
ἀξιόπιστοι Pythagoreans who attributed the “ἱερὸν λόγος not.to Pythagoras but to Telau-
ges; cf Buseb. Fist, ecel. 6.19.8), This sentence, however, shows verbal coincidence with
1}.}.. δ. 14 (below, ch. ΠῚ 1).
1. Source Problems
a third source is to be recognized, of the “handbook” type, whose
presence can be detected in coincidences between Diogenes Laertius
and the “handbook” segments of Porphyry. Actually, there is no need
to restrict lamblichus to three books; it was more convenient, if he
wanted to write on music (ch. 26), to find something in Nicomachus’
book on music than to seek out an appropriate passage in his life of
Pythagoras.6 And Iamblichus could have read personally, as Stobaeus
did, the Πυθαγορικαὶ ἀποφάσεις of Aristoxenus.!”
Thus our analysis leads, in the first instance, mainly to sources of the
first or second century: Moderatus lived under Nero,!* Apollonius
under Domitian, and Nicomachus is dated by the fact that Apuleius
translated his Introduction to Arithmetic into Latin}® Yet these neo-
Pythagoreans are themselves only intermediaries. Nicomachus cited
his authorities meticulously, Apollonius did not. So Rohde judges
Nicomachus very favorably: he is an intelligent compiler, and no
(orger. Toward Apollonius he is very negative: “‘one is best advised
not to believe anything he says.’’2° But this judgement can only be
reckoned as true a parte potiori. Apollonius too used good sources,
and it is precisely his material that has provided most opportunities for
those who, from time to time, have tried to discover really ancient
lore.2?, Nicomachus, on the other hand, who calls himself a Pytha-
yorcan, is so intimately concerned in his narrative that, at least in
sclection, arrangement, and interpretation, his personal contribution
must not be underestimated.
Vor the carlier stages of the tradition Diogenes Laertius is particu-
larly important, though his work is hard to analyze. He has woven
topether material from various handbooks, and his “card-file” method
inakes it almost impossible to discern connections of any larger ele-
tnents, Still, there is an example of parallel tradition in the Pythagoras
tun. VP iisff=Nicom. Ench, 6f. Rohde (Q 126f, 146f) assumes that Nicomachus
has! an identical passage in his life of Pythagoras.
The excerpts from the Πυθαγορικαὶ ἀποφάσεις are quite distinctive in style (lam. VP
ln 102, 174-176, 180-183, 200-213, 230-233). In addition, at the beginning of the first
και ρὲ the title is expressly cited (101). The Stobaeus excerpts are frr. 34-37, 39-41
Wehrli. Phe collection in DK 58D is somewhat helter-skelter.
mot W. Capelle, RE XV 2318-2320. Plutarch mentions Moderatus at Quaest. conv.
Η ) 1.
i Ct BE. Robbins, in M. L. D’Ooge (tr.), Nicomachus of Gerasa, Introduction to Arith-
metic, with Studies in Greek Arithmetic by F. E. Robbins, L. C. Karpinski (New York,
126) pp. 71. The peculiar RE article by F. Khem (XVII 463f) deals only with Nico-
ται παν Arithmetic and (briefly) the Theologumena, completely omitting consideration of
the extant Harmonicum Enchiridium and the largely reconstructable life of Pythagoras.—
Poche thought he was a reincarnation of Nicomachus (Marinus V. Procl. 28). This may
place Nicomachuy’ death in tg6 ap. (J. Μ, Dillon, CR τῷ [1969] 274).
et) 192:
" Nelow, ἢ. 97.
I. PYTHAGORAS IN THE EARLIEST TRADITION
article of the Suda and a Platonic scholium (Rep. 600b) which is almost
identical with it; both are referred to Hesychius of Miletus. The same
exposition is discernible in Diogenes Laertius, though broken up by
numerous insertions. So one of the main sources can be distinguished,
even if its name is unknown.”
The work of Neanthes of Cyzicus must have been an important
intermediary source.* He brought together several versions of the
origin of Pythagoras, including that of Aristoxenus;4 like Hesychius
he names Pherecydes and Hermodamas as Pythagoras’ teachers, and
enumerates the brothers of Pythagoras in the same terms as the source
common to Diogenes Laertius and Hesychius.”> Thus in Neanthes we
find the earliest example of the handbook provided with source cita-
tions and variants. Perhaps it was the one that set the trend;?¢ in any
case it furnished an exposition that was widely used—by the source of
Diogenes Laertius, by Nicomachus-Iamblichus, by Clement and
Porphyry. It is striking that Neanthes is often named along with Hippo-
botus.2”? Obviously one had cited the other; probably Hippobotus, who
wrote On Sects (D.L. 1.19), made use of the work of Neanthes,
whose special concern was with the “mythical.” Thus the sequence
Neanthes—Hippobotus may be one link in the handbook tradition.
Callimachus’ student Hermippus wrote several influential books on
Pythagoras; Josephus calls him the “most distinguished” of the bio-
graphers of Pythagoras.?® The fragments we have contain the most
22 Cf. Delatte, Vie 9-63, where the older literature is discussed. ‘‘Handbooks”’ are
responsible for the coincidences between Diogenes Laertius and Clement, Hippolytus,
Porphyry, and Iamblichus. (On this, cf. Jager passim.)
23 FGrHist 84F26-33. The book in question is the fifth book of his υθικά (F29). This
Neanthes is probably to be dated about 200 B.c., and distinguished from a historian of the
samie name of about 100 years earlier (Jacoby, FGrHist Hc, 144; for the early date, von
Fritz, Pol, 6).
24 F29= Por. VP 1-2; the emendation of Κλεάνθης to Νεάνθης is guaranteed by Clem.
Al. Strom. 1.62.2; Aristox. fr. 11a~b; Neanthes himself believed that Pythagoras came from
Syria—the most unusual view; he must, however, have mentioned the canonical version,
that he came from Samos, and in that case he must have had three versions side by side,
as do Por. VP 1 and Clement, loc. cit.
2 Εὔνοστος in Neanthes (Por. VP 2) and Diogenes Antonius (Por. VP 10) and Evvopos
in 1).L. 8.2 and Hesychius can only be secondary variants.
6 Lévy, Sonurces (cf. above, ἢ. 6) also attributes 1D.L. 8.39° Hesychius to Neanthes,
comparing Neanthes F30(- Por. VP ς 4). Here Neanthes also is harking back to Aristo-
xenus (fr. 18 Tam. VP 249).
27 Nicom. (Por. VP 61 Tam. VP τυ Neanthes F31), Uh. ar. §2.8f ( Neanthes
F330 Aristox. fr. 12). Hippobotus is named along with Neanthes for differing versions
in Clem. Al. Strom, 1.62.2; that of Hippobotus (Pythagoras a Samian, attributed by I.E.
8.1 to Hermippus) may have been also in Neanthes (above, n. 24). At DLL. 8.72, also,
Neanthes (F28) follows Hippobotus,
Ap. 1.16]. This is a somewhat tendentious passage, for Josephus found in Hermippus
indications of the dependence of Pythagoras on the Jews, The fragments are collected in
Π ΠΟ TU aif.
1. Source Problems
eccentric material in the whole Pythagorean tradition; Rohde con-
sidered the book ‘‘a malicious satire on Pythagoras.’*® But, although
it can be shown occasionally that Hermippus distorts older source
material in a rationalizing and ironical spirit,2° still it cannot all be
arbitrary invention. A student of Callimachus will, with one degree or
another of seriousness, be engaged in collecting precisely that which
is antique and odd, so that Hermippus too may have some accurately
preserved material.
Finally, the most important sources to which analysis can lead us,
because they are the oldest, are Heraclides Ponticus, Aristoxenus, and
Iicacarchus, along with Aristotle. In their case, significant chapters
can be discerned, documented by direct quotation; attempts to enlarge
the evidence by building up more complicated reconstructions hardly
yct beyond the stage of hypothesis.
The greatest unknown is the historian Timaeus. He is cited by name
2” () 111. Especially bizarre is the report that, in a war between Acragas and Syracuse,
l'ythagoras lost his life in the course of a retreat, because he refused to run through a field
οὐ beans, (The same motif in Neanthes FGrHist 84F31=lIam. VP 189ff, in the story of
Myllias and Timycha.)
0 Hermippus (D.L. 8.69) gives a rationalistic interpretation of Heraclides’ Περὶ τῆς
ἄπνου (fre. 768 Wehrli), namely that Empedocles did not revive a dead woman, but
cured a woman of Acragas named Panthea whom the doctors had given up.”
“ The dating of Epimenides to the time of Solon (contra Pl. Leg. 642d) had been re-
yarded as an unfounded invention of Hermippus, until it was corroborated by Arist. Ath.
Pol, 1 (cf. Diels, SBBin 1891, 387ff).
Md Arist, frr. tg0ff; cf. above ch. 1 2, and below, ch. II 2, 4, 6. Heraclides names Pytha-
poras in frr. 40-41, 44, 87-89 Wehrli.—Aristox. frr. 2, 11-41, 43, 47-48, 50; Dicaearchus
lst. 33-36, 41. Attempts at reconstruction have been directed mainly toward the Abaris
of Heraclides (cf. Wehrli, pp. 84ff). It has been supposed that two important scenes can
le regained. (1) The meeting of Abaris and Pythagoras, lam. VP 91-93 (Corssen, RhM
iyt2, 38f; A. Rehm, ibid. 423f; Lévy, Sources 35, Lég. 48; Boyancé, REA 1934, added
lun. VP 215ff; cf. above, n. 13). The connection of the two wonder-workers may,
however, be earlier; it is difficult to equate the divine Pythagoras of the encounter with
Abaris with the Pythagoras who in frr. 87-88 espouses a definitely human brand of
φιλοσοφία. (Rehm’s assumption, 423f, that in different works Heraclides depicted different
lythagoras types is a counsel of desperation; the depiction of the death of Empedocles,
ty. 83, seems to show that Heraclides liked to clothe the divine in a mist of mystery, so
that he can scarcely have imagined Pythagoras as simply the Hyperborean Apollo.)—
(9) A katabasis or descent to the underworld by Pythagoras, reflected in Hieronymus fr. 42
(1).1.. 8.21) and Hermippus (D.L. 8.41) (Diels, AGP 1890, 469; Corssen, RhM 1912, 22,
Kt esp. Lévy, Sources 26f, 34ff, Lég. 79ff; for an older source, Norden, Vergil VI, 35 n. 1;
Lelatte, Litt, 109). Heraclides fr. 75 is corrupt; and Wehrli’s conjecture of τὸ δεύτερον for
1 δένδρον would destroy the foundation of the attempt to establish a connection with
i icronymus (Wehrli p. 86). Since Heraclides locates the next world in the realm of the
stats, he cannot be behind the literally understood word κατάβασις in Hermippus (cf.
below, ch. IV 4). The mention of Abaris as flying, in fr. 510, is explicitly assigned to the
hook On Justice (and there is no justification for identifying this work, on this account,
with the Abaris: ef. Wehrli, 84, contra Boyancé, REA 1934, 322ff). Also, no one could
have guessed, without the explicit attribution, that the Pythagoras anecdote in ἔστ, 87-88
was from the Περὶ τῆς ἅπνου.
II. PYTHAGORAS IN THE EARLIEST TRADITION
fora series of very specific reports about Pythagoras,* so that, a fortiori
we must assume that he gave a gencral exposition of Pythagoreanism.
He apparently showed a decided sympathy with Pythagoreanism,
rooted in western Greek local patriotism.*+ Suspected of holding
Timaeus material are Justin,?> Diodorus,** and especially Apollonius.®?
In principle, the task is to define a kind of Vulgate tradition, not derived
from either Aristoxenus or Dicaearchus. Delatte and Rostagni pro-
duced, independently, bold reconstructions, not without contradicting
each other, particularly in the decisive matter of chronology. These
constructs have, rightly, been subjected to sharp criticism. It cannot be
proved that everything in Iamblichus and the other late accounts goes
back to an authority of the fourth century B.c., nor that Timaeus
would be the only eligible candidate for this position. There are an
33 FGrHist §66F13, 14, 16, 17, 131, 132.
34 One might conjecture that Timaeus was first to trace the origin of the name Magna
Graecia (Μεγάλη ‘EAAds) to the influence of Pythagoras (Nicom. in Iam. VP 30=Por.
VP 20, Iam. 166, Cic. Tusc. 5.10, Lael. 13).
35 20.4. For the attribution to Timaeus, cf. A. Enmann, Untersuchungen tiber die Quellen
des Pompeius Trogus fiir die griechische und sicilische Geschichte (Dorpat, 1880). Justin 18 is
equivalent to Timaeus F131, and Justin 11 corresponds to Iam. VP $6, which directly
follows a fragment of Timaeus (F17). The different reports about the luxurious living of
the people of Croton (Tim. F44-45. Justin 1-2) can be reconciled (von Fritz Pol. 46f.)
Still, this does not prove that Timaeus is Justin’s only source. Is the impression of the
unity of Justin’s exposition perhaps only the result of its severe compression, and therefore
to be credited to the epitomator rather than to the unitary source? Pompeius Trogus did
use a number of different sources (O. Seel, Die Praefatio des Pompeius Trogus [Erlangen
1955] 18ff).
36 19.3-12. Diodorus is following a highly rhetorical, moralizing source, which treats
the Seven Sages in the same manner (9.1-15; Schwartz, RE V 678f). It was later than
Callimachus (10.6.4). Among its sources was Aristoxenus (Schwartz 679; von Fritz,
Pol. 22ff), but there is no distinctive trace of Timaeus. The relation of the Damon-
Phintias story in Aristoxenus (fr. 31) to the version in Diodorus (10.4.3) is controversial,
in Aristoxenus, Dionysius wishes to test the steadfastness of a Pythagorean in danger of
death, and therefore condemns the innocent Phintias, whereas in Diodorus the latter had
really planned an attempt on the tyrant. Wehrli (Aristoxenus p. 57) finds that Diodorus’
version, since it is simpler, is the earlier; but Cobet (Collectanea critica [Leiden, 1878] 433)
and Nauck (LVII 40) stressed that the famous dénouement, Dionysius’ wish to be admitted
as a third member of their friendship, only makes sense if there was no deadly enmity
but only a test engineered by the tyrant. Thus Aristoxenus’ account is after all the original,
and Diodorus is not using pre-Aristoxenian material.
87 In the speeches of Pythagoras, Iam. VP 37-56, 56 corresponds to Timeaus F17; there
is a relationship between 42 and Timacus F146; 56 agrees with Justin 20.4.11, and 37, 40,
and 47 are related to 1D.L. 8.22f. Local tradition of south Italy is evident in 40, 44, 46, 50,
and $2. Furthermore, Timacus F13 corresponds to lam. VP 71-72 (von Fritz, Pol. 39). It
is inferred from the connection with Timacus that lam. VP 71-72 comes from Apollonius,
Rohde, Q 137; Delatte, Litt, 856 (The latter adds sections 74 -78 and therefore misinterprets
the Lysis letter; cf. Burkert, Phifologus 1961.) ‘Timacus F147 is to be compared with Tam.
VP 45. On the basis of these correspondences, the conclusions are drawn that (1) the
speeches of Pythagoras in Tam. VP 47-$7 are taken entire from ‘Timacus (“selon toute
vraisemblance textuellement Timaios,” Boyancé, ReePH 29 {144} 181), and chat (2) all
1. Source Problems
uncomfortable number of intermediate sources, passing the material
on ina continuous process of variation, combination, and compilation;
and other historians, like Theopompus*® and Duris,3® also made
mention of Pythagoras. Also, it would be strange if Antiochus of
Syracuse, Philistus, and Ephorus had nothing to say about Pythagor-
cans in connection with the history of Magna Graecia; but not the
slightest trace of such is discernible.
We cannot divide the whole of the later tradition neatly among the
fourth-century sources. On the whole, the “later”’ tradition seems
to be not so much the result of unscrupulous falsification as of simple-
minded, naive compilation and transmission of whatever could be
found, contradictions and all. Still, even if the source analysis takes us
back as far as we could hope, to the very beginning of biographical
study in the early Peripatos, Zeller’s skepticism cannot be considered
wther Apollonius sections have this origin, viz. Iam, VP 2158 (Delatte, Litt. 25 τ. 3;
ilimapreeing, Boyancé; cf. above ἢ. 13), and especially Iam. VP 254ff (without hesitation
Nemtermann 37ff; Delatte, “1.4 chronologie pythagoricienne de Timée,”” Musée Belge
iat, §-13: “Apollonius se révéle tributaire de Timée pour son histoire du Pythagorisme
presque toute entiére” [13]; cf. Litt. 86, Vie 169, Rostagni, “Pitagora e Pitagorici in
Timeo,” SerMin IE 1, 3-50). On the other hand, von Fritz, Pol. 55ff. shows that in Iam.
|} 254ff Apollonius’ method is that of ‘‘ what might almost be called large-scale historical
trencay painting” (61), that he uses much old material but cannot be relied on for details.
('t. Jacoby, FGrHist IIb, Komm. 550ff, Noten p. 325 n. 191c. The ambivalence of the
aruutmients is shown, for example, by the repeated connection of Pythagoras with
Vatromenium (lam. VP 33=Por. VP 21, from Nicomachus; Iam. 112, 136)—but
‘Vattromenium was not founded till 403 B.c. Shall we say that the local patriotism of the
ἐμ! ian from. Tauromenium isin evidence here, or that Timaeus, ofall people, is excluded
ly such a monstrous anachronism?
OPC er Hist 11§F72-73.
Wuris FGrHist 76F22-23; here Samian patriotism plays a part.
“The catalogue of Pythagoreans Iam. VP 267=DK 58A is a quite isolated piece of
ettelition: 145 of its 23 5 mames occur only here (cf. the lists of Brown, rvff). And it is not
juinblichus’ own composition, since he elsewhere names 18 other Pythagoreans. It is
imlependent of the ps.-Pythagorean literature, lacking 18 of the “Pythagorean” writers
ἐπ] μας by Thesleff. Most surprising is the omission of the Locrian Timaeus; Diels’s
μ΄ ρα] to include the Parian Timaeus among the Locrians (DK I 447.3) is arbitrary.
(Iere is without doubt a close relation to Aristoxenus (Rohde, Q 171): compare the
Vhliasians p. 146.6 with Aristox. fr. 18-19; Damon and Phintias, p. 146.1, with Aristox.
ty qt; Flippo asa Samian, p. 146.3, with Aristox. fr. 21; Philolaus and Eurytus as Taren~
tines, py. 144.01, with Aristox. fr. 19; Charondas and Zaleucus, pp. 146.12, 145.8 with
Avwtox. fr. 43.—The form of the list of names, classified according to geography, corre-
sjroncly τὸ the documentary tradition which we see in inscriptions (e.g., IG II? 1697ff,
ty 4qolt), not τὸ the literary tradition (e.g. lam. VP 266, D.L. 10.22ff, Por. VP 7-9, Regen-
μου, RE s.v. Pinax, XX 1451f). There seems to be genuinely Italian material among
the tames; of A. von Blumenthal, Glotta 17 (1929) 104f, 152ff. They take us down to the
fat halfofthe fourth century (cf. also Oldfather, RE X174). The only possible candidate
tin authorship seems to be Aristoxenus himself, working in the documentary method of
the earhest Peripatos (οὗ PGri fist 124P23), Philochorus (PGr# list 328T1) wrote a Συναγωγὴ
ἡμιίδων ἥτοι Πυθαγορείων γυναικῶν. Is there perhaps a relation between this and the
accond part of the catalogue?
II. PYTHAGORAS IN THE EARLIEST TRADITION
refuted.*! It is precisely with the authors in the circle of Plato and
Aristotle that the real problems begin.
Most obvious is the contradiction between Aristoxenus and
Dicaearchus, regarding the catastrophe that overwhelmed the Pytha-
gorean society. One of the two reports must be basically wrong:
either Pythagoras withdrew to Mctapontum before the outbreak of
the unrest and died there (as Aristoxenus says) or he and his followers
were hounded from city to city (as Dicaearchus has it). Like his
doctrines, the life of Pythagoras also becomes a mirror image of real
controversies in the schools.48 On the one hand there is the controversy
over the primacy of the theoretical or the practical life (Bios θεωρητικός,
βίος πρακτικός). In this respect Heraclides thinks of Pythagoras as the
apostle of pure “theory”; to Aristoxenus and Dicaearchus he is an
active politician.44 Then, in the matter of his doctrines on the soul,
Heraclides, following Plato, advocates the immortality of the soul and
secks proofs for its independence of the body. This explains his interest
in Abaris, in the metempsychoses of Pythagoras, and the “apparently
dead woman” of Empedocles. Like Kenocrates, he expands Plato’s
myths into a system that comprehends both science and religion, and
joins together astronomy and immortality.4* We need not be bound
by Timaeus’ characterization of him as a παραδοξολόγος ;* but in his
treatment of the anecdote about φιλοσοφία we can show that, contrary
to history, he projected Plato’s conception onto Pythagoras.*”
Dicacarchus said that “soul” is a mere word (frr. 7ff), and there is
an unmistakable irony in his account of Pythagoras’ doctrine of
metempsychosis. Pythagoras was once, he says, a beautiful courtesan.*
Dicaearchus has Pythagoras, at the end, stumbling from one catastrophe
to another; and the Locrians, who were famous for their εὐνομία,
denied him admittance to their city.*®
Aristoxenus dealt most fully with Pythagoras and his pupils. He was
41 Cf. the introduction, above. Not only Capparelli (I 351 and passim) but also Morri-
son (CQ 1956, 135) maintain that Zeller’s skepticism can no longer be upheld in view
of the results of source analysis.
4 Below, ch. II 2.
43 Cf. the discussion of Wehrli in his commentaries on the fragments of the Peripatetics.
44 Jaeger, SBBlu 1928, 396 n. 1, 4158, after Rohde, KiSchr (L110.
4 Below, ch. IV 4.
4 PGrHist §66F6 Weraclides fr. 84.
47 Burkert, Hermes 1960. 1S9ff.
45 Er, 36. The irony was noticed by Rohde, Psycte TH App. to (Eng. tr), and Wehrli,
p52.
Br. 44 14. Here there is polemic against Aristoxemus (fr. 43), who made Zaleucus a
pupil of Pythagoras,
1. Source Problems
ἃ native of Tarentum, and cited in evidence the acquaintance of his
father with Archytas,®° as well as his own acquaintance with the “last”’
of the Pythagoreans.*! Clearly, he put himself forward as an expert in
l'ythagorean matters, just as he was, at the same time, an authority in
musical theory. He considered false, however, the numerical theory of
the tone intervals, which is specifically attested by Plato and Aristotle
as Pythagorean,® and did not believe in the immortality of the soul,
which he called a ἁρμονία, with all the consequences drawn from that
idea in Plato’s Phaedo.® Thus Aristoxenus has to steer a complicated
course.4
In his Harmonics, Aristoxenus disposes in a single sentence of the
music theory advocated by the Pythagoreans down to Archytas,
without even naming the Pythagoreans;55 what does not suit him is not
mentioned. Aristoxenus characterizes the Pythagoreans with whom he
was acquainted, as the “last,” suggesting that with them the school
cuded—but there was a Pythagorean named Lycon who wrote against
Aristotle, and there are also other traces that lead into the later fourth
century.55> Aparently Aristoxenus does not recognize these men as
Pythagoreans. Finally, the assertion that Pythagoras was fonder of
beans than anything else can only be veiled polemic against the taboo
on beans attested by Aristotle and Heraclides.*”
The Πυθαγορικαὶ ἀποφάσεις show the lineaments of a rational, clearly
articulated ethic, oriented toward practical needs. Its political precepts
are surprisingly similar to those of Plato’s Republic, though the pre-
decessors of Plato named by Aristotle in this connection are Phaleas
Tr. 30; cf. ὅτ. 47ff; Plut. De gen. 592: on fr. 2, von Fritz, Gnomon 32 (1960) 495.
"hry, 18-19; cf. fr. 1; below, ch. II 6.
δ Below, ch. V 1.
Err. α1 88.
‘4 Aristoxenus is the fullest of the ancient sources for Pythagoreanism, and therefore
(he question of his credibility is especially important and much discussed. The very
rationality which characterises Aristoxenus’ Pythagoreans, far from mysticism and magic,
weins a favorable sign to some scholars, suspicious to others, On the positive side, follow-
ing, Krische, are, for example, Delatte (Litt. 19. Pol. 213), Rostagni (ScrMin I 137ff),
Hlowald (JAW 197 [1923] 163), with qualifications von Fritz (Pol. 27ff), Wuilleumier
hesitutingly (6o2ff), On the negative side: Lévy (Sources 44ff), Frank (260, AJP 1943, 221
{where he wrongly depends on the formulation of Euseb. Praep. evang. 1§.2=Aristox.
ty. og, followed in this by van der Waerden, SA 107]).
© Paye 12 Μ,, emphasized by Frank, AJP 1943, 221f. It should be kept in mind, in this
connection, that, according to Aristocles in Eusebius, Aristoxenus makes malicious
icterence to Aristotle without calling him by name (fr. 64; cf. Wehrli. p. 68).
On this, below, ch. ID ὁ.
δ᾽ Below, ch. ΠῚ 4 αν 124. The report on Pythagoras as an cater of meat in fr. 25 is biased
ithe same direction; cf below, cho U4, no cis.
Π. PYTHAGORAS IN THE EARLIEST TRADITION
and Hippodamus, and not the Pythagoreans.5® When Aristotle says
that none of the “ancients” had dealt in detail with the nature of Τύχη,
whereas Aristoxenus presents as “Pythagorean sayings’ precisely
what the Eudemian Ethics expounds as réyn,°® then it is obvious that
fourth-century ethics, and not ancient Pythagorean tradition, is being
set forth. Finally, Zaleucus and Charondas, contrary to any chrono-
logical scheme, are made into pupils of Pythagoras,®° and we learn that
Pythagoras was the first to introduce weights and measures into
Greece.®!
Not that Aristoxenus invented it all; he had sources of information,
but like the Platonists he interpreted Pythagoreanism.in accordance
with his own preconceptions.® And this is the basic fact that finally
emerges from analysis of the sources: though the late tradition may be
traced back to the writers of the fourth century B.c., this does not lead
us onto firm ground but into the precarious territory of that day’s
controversies. Instead of reliable facts one finds the shifting claims to a
Pythagorean tradition apparently already in a state of flux, for the
benefit of the peculiar concerns of the pupils of Plato and Aristotle.
One may attempt to discover historical truth from amidst the polemic—
not, naturally, isolating the individual testimonies, but considering
each along with the others that contradict it. But the foundation of
historical research must be the pre-Platonic evidence because this
belongs to a time before Pythagoreanism sank into the quicksand of
58 Cf. A. Rivaud, “‘Platon et la ‘politique pythagoricienne,’” Mél. Glotz (Paris, 1932)
779-792: “‘Aristoxéne a reconstruit, avec le secours des textes de la République, une
politique pythagoricienne qu’aucun auteur pythagoricien n’avait jamais formulée
expressément”’ (784). M. Pohlenz (NGG, ph.-h. K],. 1924, 19-32) inferred as the source of
the ps.-Demosthenic first oration Against Aristogeiton a Pythagorean writing Περὶ νόμων.
But aside from the problems of detail involved in such a reconstruction, it would not
lead us to pre-Platonic terrain. The trial of Aristogeiton took place about 324 B.c., so that
there might be influences from Socratics, Academics, or Peripatetics, not even excluding
Aristoxenus.
59 Aristox. fr. 41; EE 1246b37ff, 1248b2, 1214a15; the testimony of Aristotle about the
“ancients,” Phys. 195b35ff. But compare 196b5; H. Tager, De Aristoxeni libro Pythagorico
(Diss. Géttingen, 1922). This unpublished dissertation has unfortunately remained almost
unknown; even Wehrli does not use it. Yet Wehrli too says (50): ‘The hallmark of the
᾿Αποφάσεις is to lay claim to Academic-Peripatetic material on behalf of the Pythag-
oreans.”’
60 Fr, 43; followed by Posidonius (Sen. Πρ. go.6).
61 Pr, 24; cf. below, ch. VI τ.
® The Πυθαγορικαὶ ἀποφάσεις seem to be a modern substitute for the acusmata, Wehrli
(61) refers, on fr. 37, to Arist. Met. o85b30. Concise formulations of popular “ wisdom”
- καιρός 7, ὡρίζοντο yap ἐπιπολαίως (987022) are analyzed and more fully developed
by Aristoxcnus.
2. Historical Background
the school controversics.6 We may reasonably expect information
about the early school from those later authors who are critical of
Pythagoreanism rather than from those who identify themselves with
1 and are therefore forced into radical reinterpretations of it. From this
point of view the testimony of Aristotle again emerges as especially
important, for he clearly, in his lost book on the Pythagoreans, was
concerned to collect material, rather than to interpret it in such a way
as to fit into his own system.
It is customary to complain about the sparseness of early testimonies.
Stull, whereas we have no explicit reference to Anaximander or Par-
menides by any fifth-century author, there is a quite imposing array
of references to Pythagoras. Interestingly, most of these had already
len assembled in ancient times; we have them as quotations in the
lives of Pythagoras.®* The history of Pythagoreanism was already, at
that time, the laborious reconstruction of something lost and gone.
2. HISTORICAL BACKGROUND
It is only in post-Aristotelian sources that biographical and historical
details regarding Pythagoras and the Pythagoreans are to be found.
‘Ihe most important accounts are those of Aristoxenus and Dicaearchus,
lit they differ diametrically on one vital point—the role of Pythagoras
in the revolution in Croton. Considering these authorities and also the
lucr tradition, of which some indeterminable portion comes from
‘Timacus, one is tempted to say that there is not a single detail in the life
of Pythagoras that stands uncontradicted. It is possible, from a more or
"Ina still more radical spirit, Gigon (Ursprung 123f) bases his account exclusively on
(itth-century testimonies, and Frankel (DPh 354ff) relies entirely on Xenophanes, Hera-
tlitus, and Empedocles. The most ancient evidence is collected in Zeller SBBin 1889
uks φῇ KISchr I 458-472, Frank 356f, Timpanaro Cardini 12ff.
“41 summary: Xenophanes DK 21B7=D.L. 8.36; Heraclitus Bgo=D.L. 9.1, B81=
"14. Rhet. Lc. $7; 62-}-Schol. Eur. Hec. 131, B129=D.L. 8.6; Empedocles B129=Tim-
wus UGrHlist $66F14—=D.L. 8.54=Nicom. (Por. VP 30=Iam. VP 67); Hdt. 2.81, 4.95,
it. Hellanicus FGrHist 4F73; lon of Chios DK 36B2=D.L. 8.8, Clem. Al. Strom. 1.131,
ty 1.1. 1.120; Glaucus of Rhegium D.L. 9.38; Democritus A33=D.L. 9.38; DK
ΝΠ οι Dissoi ἰοσοί 6.8. Contemporary with Plato: Isoc. Bus. 28; Alcidamas D.L. 8.56;
Akidamas, Arist. Rhet. 1398b9; Anaximand. Hist. FGrHist 9T1= Suda s.v.; Andron
af Ephesus D.L. t.rrg, Euseb. Praep. evang. 10.3.6 (DK 7A6); Theopompus FGrHist
risty2 73. On Antisthenes, see below, ch. II 2, n. 38.
™ Most discussed in antiquity were the verses of Empedocles, B129 (cf. the previous
note), DAL. 8 adds Xenophanes, Heraclitus B129, lon B2. Clement dates Pythagoras by
Heraclitus 540 (Strom, t.12y.4). Herodotus’ story of Zalmoxis influenced the later tradi-
Hew inanany ways (cf DOE. 8.2, Por, UP ἀ4}, lam. VP 173).
II. PYTHAGORAS IN THE EARLIEST TRADITION
less critical selection of the data, to construct a plausible account;! but
it is bound to rest on shaky foundations, for no documentary evidence
has appeared.?
A mainstay in discussion of the chronology’ is the report of Aristox-
enus, according to whom Pythagoras left Samos for Italy in 532/531
B.C. because of the oppressive tyranny of Polycrates.4 On the other
hand, Eratosthenes identified the philosopher Pythagoras with the
Pythagoras of Samos who was an Olympic victor in 588 B.c.® The
1Cf. Zeller I 380ff. The most important of the older studies is Krische. See further
Delatte, Pol.; von Fritz, Pol. and RE; Minar; Ciaceri II 86ff, 2178; Dunbabin 35off,
366ff, 360ff.
2 A striking piece of original evidence for the extraordinary fame that surrounded
Pythagoras as early as the 5th century is perhaps to be seen in the ‘‘Pythagores” coins of
Abdera. See Seltman, Greek Coins (London, 19331, 1955?) 142ff, pl. xxviii 11 (cf. Numis-
matic Chronicle 6.9 [1949] 21); J. M. F. May, The Coinage of Abdera (London, 1966) pp.
157, 167, with pl. 13 no, 218 (P183); also, addendum p. 176 with text-cut and description
of no. 218/r (P183/1). The date is given as 430/420 B.c. On the obverse the coins of Abdera
show the picture of a griffin, after that of the mother-city Teos. The reverse has, after
about 450 B.c., the name of the mintmaster in charge (who may have been the priest of
Apollo, Abdera’s patron deity), and in the quadratum incusum a symbol which changes
from one mintmaster to another and, at least in some cases, bears an obvious relationship
to his name—as when Python uses a tripod, Euagon a prize amphora, Nicostratus a
soldier, or Molpagores a girl dancing (nos. 345-348, 293-295, 313-315 May). The coins
of the mintmaster TY@ATOPH®S display an idealized, bearded head, in two different
types. The design must be related to the person of the mintmaster, and may be related to
his name. Seltman’s bold conclusion, that the person represented is the famous Pythagoras
himself, seems likely. G. M. A. Richter’s suggestion that the mintmaster portrayed himself
(Greek Portraits IV [Brussels, 1962] 17-19) has been disproved bythe appearance of a second,
different type. The representation must be that of an ideal person, not a living individual;
see W. Schwabacher in Stockholm Studies in Class. Archaeology § (1968) 59-63. In any case,
we know Pythagoras was known in Abdera at this epoch (Democritus, D.L. 9.38;
Herodotus cites the Greeks living on the Black Sea and Hellespont for the relation between
Pythagoras and Zalmoxis; cf. below, ch. II 3). To be sure, the interpretation cannot be
proven, and there are other coins of Abdera with heads that cannot be identified.—
Certain ancient portraits have been thought to depict Pythagoras: (1) a bust from Her-
culaneum, the copy of a non-Attic work of the 4th century B.c. (K. Schefold, Die Bildnisse
der antiken Dichter, Redner und Denker [Basel, 1943] τοσοῦ); (2) a neoclassical statue in Rome
(Schefold 160.1); (3) a bust in Ostia (G. Becatti, “Ritratto di un vate antico,” Bolletine
d’ Arte 34 [1939] 97-110). In addition, Pythagoras appears on coins of Samos from Trajan
to Decius (Head 606; Schefold 173.17-18; British Museum Coins, Ionia, nos. 237, 257, 287,
351, 365), and ona contorniate from the 4th century A.p. (Schefold 173.19).
3 Cf. Zeller I 381 n. τ; Rohde, Q 118ff; Jacoby, Apollodor 215ff; Lévy, Sources 2¢¥;
von Fritz, Pol. 68ff and RE 179-185.
4 Aristox. fr. 16 W., followed by Apollod. FGrHist 244F338-339; cf. IIb Noten p.
326 ἢ. 198. In spite of this, Aristoxenus made Zaleucus and Charondas pupils of
Pythagoras (fr. 43).
® PGrHist 241 Ἐπ: DVL, 8.47; ibid. 48, an epigram on this Pythagoras by Theaetetus
of Cyrene (time of Callimachus), and (49) an anonymous epigram which gives hi:
father’s name as Crates. Eratosthenes’ lise of Olympic victors must have been reliable for
the pertod in question; Le. the Olympic victor of ¢88 is a historical figure (Rohde, ©
118). Lévy (Sources 2017) sees the whole as part of the fégende denfance that: yrew up abou
Pythagoras: even as a boy he won an Olympic victory,
2. Historical Background
tradition provides examples of other early dates, too,® and in the other
direction Alcidamas and Timaeus make Empedocles the direct pupil
of Pythagoras, which would push his date far down into the fifth
century.” Xenophanes and Heraclitus refer to Pythagoras;* for
llcrodotus he does not belong to a very distant past.® A statement in
luimblichus, which probably goes back to Aristotle, places him after
the Seven Sages,!° and an old tradition makes him a pupil of Pherecydes
of Syros.4 Thus various strands of evidence lead to the second half of
the sixth century and to this extent support Aristoxenus, even though
there is in his reference to Polycrates a certain tendentious quality.
Pythagoras came from Samos,! and his father’s name was Mnesar-
chus.'3 We cannot determine the age of certain local traditions of Samos
"Pliny the Elder (HN 2.37) dates an astronomical discovery of Pythagoras to 612/609;
atul at 36.71 he places him in the reign of Psemetnepserphres (= Psammetichus I, 664-610
n.¢., or Psamimetichus II, 594-89; naturally the two could easily be confused).—Eusebius
(lr. Chron. p. 14.30 Karst) places Pythagoras under the successor of Sanherib (i.e. after
O41 1.0.3 cf Jacoby, FGrHist Ia 296, on 273F79: not Alexander Polyhistor, but an addi-
tran by Eusebius). The Numa story (Jacoby, Apollodor 225f; cf. Burkert, Philologus 1961)
and the encounter with Phalaris lam. VP 215ff) were invented without consideration of
ihtonological matters.
* Alcidamas ap. D.L. 8.56, Timaeus FGrHist 566F14=D.L. 8.54. There is no reason to
doube the text, as Levy does at Sources 44. n. 3; neither the dating of Pythagoras nor that
of Empedocles by Timaeus is known. (For contradictory reconstructions, see Delatte,
Afusce Belge 1920; Rostagni, SerMin II 1 pp. 3-50; cf. von Fritz, Pol. 47ff, RE 180-184;
Jacoby FPGrHist Ib Komm. 552.) At lam. VP 44 Pythagoras is represented as speaking of
4 Crotonian Olympic victors; and this could only apply to the years between 508 and 496
(ct. A. Moretti, Olympionikai [Rome, 1957]). Canthis come from Timaeus (cf. ch. I]1,n. 37)?
* Zeller felt (1640 n. 1; cf. Rathmann 38) that in fr. 7 Xenophanes speaks of Pythagoras
as though he were already dead; and Heraclitus (fr. 40) speaks of Pythagoras in the same
Iucath as Hesiod—though scarcely on chronological grounds. Cf. below, ch. II 6.
"4.06: cf. below, ch. II 3.
“Tam. VP 83; cf. below, ch. II 4.
"See below, ch. II 3. The synchronism of Pythagoras and Alcmaeon at Arist. Met.
wkoa2y is an interpolation; cf. above, ch. 1 2, n. 6.
"ΙΔ. 4.95; Hellanicus FGrHist 4F73; Isoc. Bus. 28; Hermippus ap. D.L. 8.1; etc.
IIe was, however, a “Tyrrhenian” according to Theopompus FGrHist 115F72, Aristox.
tt. ata-c W., and Aristotle (MSS Aristarchus) fr. 190, from one of the ‘Tyrrhenian
ishinds of the northern Aegean.” (The statement in Plut. Quaest. conv. 727b-c that he came
trom Keruria, is Plutarch’s improvisation.) He was a Syrian from Tyre according to
Neanthes PGrHist 84F29 (cf. Apollonius ap. Iam. VP sf, 7, 13). Lycus (?) ap. Por. VP ς
sys λέγουσι yap αὐτὸν οἱ μὲν εἶναι Σάμιον οἱ δὲ Φλιάσιον, οἱ δὲ Meramovrivoy (cf. below,
ch. ΠΠχ,.π..11). Phlius appears in the genealogies, varying in detail, of D.L. 8.1 and Paus.
1.13.2, (For Pythagoreans from Phlius, cf. Aristox. fr, 19 and Echecrates in Pl. Phd.; for
lythagoras and Phlius, Heraclides fr. 87 W. A Φλιάσιος σωμασκητής is distinguished as a
homonym by 1. L. 8.46; cf. below, ch. Il 4.) Zeller’s conciliatory suggestion (after K. O.
Miller) that Pythagoras came from a‘ Tyrrhenian-Pelasgian family that had migrated
from Philius to Samos” (f 380.2) will scarcely find support any more.
“11 leraclitus fr. 129, Hdt. 4.9.4. In kamblichus the name is consistently miswritten as
“Manemarchus.”” ΤῸ is given as Marmacus in the genealogy of D.L. 8.1 (cf von Fritz,
RE XXIV 172; Mamercus as son of Pythagoras, Phat. Aem, Paul 1); Demarato (or de
Muarato?) πες, Justin, 20.4.3.
II, PYTHAGORAS IN THE EARLIEST TRADITION
in which Pythagoras figured.4# As early as Herodotus he is connected
with Egypt.15 In the tales of his journeys to visit the Phoenicians, the
Chaldaeans, and the Magi there is a good deal of imaginative con-
jecture, though he unquestionably had some sort of contact with the
Orient.!® The principal site of his activity was southern Italy, and the
traditions center about the cities of Croton and Metapontum.!” It is
well attested that his death took place in the latter city.18
Not surprisingly, Pythagoras’ teaching is permeated with the kind of
religion characteristic of Magna Graecia. Typical of this is the promi-
nence of the chthonian divinities—Demeter, Persephone, Dionysus—
and of eschatological beliefs, especially the type that produced the
numerous representations of the journey of the deified dead into the
Beyond.!® This religious situation is probably older than Pythagoras.
M4 Antiphon (Por. VP 9 = Iam. VP 26f) speaks of a Πυθαγόρου καλούμενον ἔτι καὶ
viv ἡμικύκλιον, ἐν ᾧ Σάμιοι περὶ τῶν κοινῶν βουλεύονται, and an ἄντρον before the city,
Local patriotism may play a part in the report of Duris (FGrHist 76F23) about an epi-
gram and dedicatory offering of Pythagoras’ son Arimnestus in the Heraeum at Samos. A
‘“*Samian poet” is cited by Apollonius (Por. VP 2 = Iam. VP 5). On Samian coins, see
n. 2 above.
15 5.81 (below, ch. H 3); Isoc. Bus. 28; Hecataeus of Abdera FGrHist 264F25 = Diod.
1.69.4, 96.2, 98.2. The letter of introduction from Polycrates to Amasis (Antiphon ap.
Por. VP 7 = D.L. 8.3) is an invention based on Agesilaus’ letter to Nectanebis in behalf
of Eudoxus (D.L. 8.87). See further Zeller I 387.1.
16 Aristoxenus alleged that he paid a visit to “ Zaratas” (fr. 13 = Hippol. Ref. 1.2.12;
on the extent of the Aristoxenus material see Wehrli sof and W. Spoerri, REA ς7 [1955]
267-290). Alexander Polyhistor followed him in this (FGrHist 273F94; cf. Jacoby IIIa
2048). It was a chronologically impossible idea to imagine that Pythagoras got from
Egypt to Babylon as a prisoner of Cambyses (JG XIV 1297, II 20 = FGrHist 252B7;
Apul. Flor. 15, p. 6; Th. ar. 53.1ff; lam. VP 19; the episode is modeled after the Demo-
cedes story). Further references in Zeller I 384ff, as also for his supposed contacts with the
Thracians (Zalmoxis, below, ch. II 3, n. 202), with Arabs, Jews, Indians, and the Druids
of Gaul. The god Men and the taboo on white roosters are certainly from Asia Minor
(ch. II 4, ἢ. 47), and the wearing of trousers is Persian or Scythian (Ael. ΝῊ 12.32).
17 Alcidamas in Arist. Rhet. 1398b10f: ᾿Ιταλιῶται Πυθαγόραν (τετιμήκασιν). The
legends are mainly localized in Croton (of which Caulonia was a colony) and Metapontum
(ch. Il 3). Pythagoras is Metapontine to Lycus (?) ap. Por. VP 5; Brotinus (DK 17.1),
Theano (lam. VP 267, pp. 146.22 and 132), and Hippasus (DK 18.1-2) are sometimes from
Croton, sometimes from Metapontum.
8 Arist. fr. 191, Aristox. fr. 18, Dicaearchus fr. 34-35 (cf., for further material, Zeller
I 417ff; Delatte, Pol. 203ff, Vie 241ff, Lévy, Leg. 63ff). In Metapontum, Cicero saw
“Pythagorae ipsum illum: locum, ubi vitam ediderat, sedemque” (Fin. 5.4; this may of
course be a rediscovery for the benefit of tourists). According to Porphyry (VP 4 =
EGrHist §66F131), Timacus spoke of the house of Pythagoras in Croton, and the same
reportis given, but with Metapontum as the city, by Favorinus (1).1ν. 8.1), Justin (20.4.18),
and famblichus (VP 170), Val. Max. 8.15.1 is corrupt. It is more likely that Porphyry
made an error (or his source; ch Rohde, Q 133.1; Delatte, Vie 183; Lévy, Sources $4.1,
Lég. 65) than that Timacus purposely substituted Croton for Metapontum (as Jacoby,
Prd list Ib Koon. $60; Jager 2off).
Wf Giannelli, Wuilleunier 46g «60. Wilamowitz (Platow TE 84) had already called
attention to the religion of the western Greeks and ity connection with Pythagoreanism,
2. Historical Background
The prominence of Aphrodite beside Persephone in the votive tablets
from Locri?® surely cannot be derived from Pythagoreanism, but there
is nothing distinctively Pythagorean in the famous gold tablets, either.”
Pythagoras entered a religious world of a peculiar character, in which
Mediterranean, Italic, and pre-Doric, Achaean elements were amal-
gamated.”?
It is remarkable that only Croton and Metapontum, among the
south Italian cities, are noted for the worship of Apollo. There seem to
have been quite ancient Apollo cults in Metapontum,”’ as well as in
Macalla, which belonged to Croton.** The facts that Croton used the
tripod of Apollo on its oldest coins (about 550 B.c.),25 that Caulonia,
Ὁ Quagliati, Ausonia 3 (1908) 136-234; W. A. Oldfather, Philologus 69 (1910) 114-125,
RE XIE 1349ff, Delatte, Pol. 7; see now H. Priickner, Die Lokrischen Tonreliefs (Mainz
1468), who points out their connection with the ritual prostitution in the temple of
Aphrodite in Locri(Justin 21.3, Clearchus fr. 43a W.). The ancient authorities contradict
cach other on the connection between Locri and Pythagoreanism; Aristoxenus (fr. 43)
calls Zaleucus a Pythagorean, but Dicaearchus (fr. 34) tells how the Locrians refused
Pythagoras admission to the city. On “‘Timaeus of Locri”’ see above, ch. I 4, n. 7.
211K 1B17-21. A new tablet, of the fourth century, has been discovered at Pharsalus
(N. M. Verdelis, Arch. eph. 1950-1951, 98ff). Six of the total of twelve were found in
southern Italy (Petelia, Thurii). It is usual to call them “Orphic-Pythagorean” (cf.
Nilsson, II 223ff; Ziegler, RE XVIII 1386ff). On the other hand, some have claimed that
they are Pythagorean but not Orphic (H. M. R. Leopold, Mélanges d’Archéologie et
d'histoire 39 [1921-1922] 170f; Wilamowitz, GidH II 202f; Thomas 13off; cf. Cumont,
Symb, 371, n. 1, 377, n. 6, 423, n. §; Bidez, Eos 9.2). This seems less likely, however, since
the oldest of all has been discovered in Thessaly. The central figures are Persephone and
lionysus (who was worshiped in Metapontum as Dionysus Eriphius: Thomas 14off).
Kubuleus, also present, makes one think of Eleusis (Thomas 138f; cf. 146ff). Nothing
points unequivocally to Orpheus, but specifically Pythagorean elements cannot be-
demonstrated, either. There is no direct mention of metempsychosis on the tablet from
Vharsalus (or that from Petelia). Vague associations, like the occurrence of the cypress
(cf. I).L. 8.10) and the choice between right and left (cf. above, ch. I 2, n. 49; on the
"Y," the littera Pythagorae, see Burkert, Philologus 1961, 230) cannot suffice to show that
Pythagoreanism is the source of the whole. We must also consider the very old silver
tablet from Posidonia (IG XIV 665; cf. Giannelli 147f), with the inscription TAL’ ΘΕῸ
T'< A> TIAIAOZ EMI (like DK 1B18.8: δεσποίνας δ᾽ ὑπὸ κόλπον ἔδυν χθονίας βασιλείας).
Nchind this lies an essentially pre-Pythagorean mystery rite, in which Pythagoreans could
of course participate.
#4 On relations between Metapontum and Messenia, see ch. II 3, n. 242. On relations
with the Etruscans, Furtwangler, Gemmen III 202f, 254ff; Ferrero 120ff; van Essen
(skeptical). J. Gagé tries to show that Dodona was a center of religious influence for Italy
(RETR 145 [1954] 137-167; 146 [1954] 18-50, 129-139; 147 [1955] I-31).
“1 Apollo Lyceus and Apollo Carneus; cf. Giannelli 6rf.
44 Ps -Arist. Mir. ausc. 107, 840a15; cf. Giannelli 162-167.
* The coins of Croton have often been brought into connection with Pythagoreanism.
Ct. Head, οὐ (negative); Giannelli 152-153 (doubtful); Boyancé, Muses 238; Mary White,
[HS 74 (1984) 36-43; A. M. de Guadan, Numisma 34 (1958) 9-24. The most enthusiastic
wdvocate of the Pythagorean interpretation is C. Seltman, who even supposed that
Pythagoras, who was the son of a gem engraver (D.L. 8.2), designed these coins himself
(Numismatic Chronicle 6.9 [tgay] 1-21, Greek Coins [London, τς 43] 77f). This is refuted by
the face that the oldest coins of Magna Graccia were issued by Sybaris, not Croton; cf.
Ie Vogel, Pythagoras §2~§4.
II. PYTHAGORAS IN THE EARLIEST TRADITION
a colony of Croton, showed on its coins Apollo καθαρτής with a stag,”
that in about 470 Mctapontum issued Apollo coins?’—these facts
doubtless have implications about the soil in which Pythagoras’
doctrine took root. His unique success had a quite individual kind of
background, which combined the piety of chthonic mysteries with
worship of Apollo the “purifier.”
The traditions about the persons surrounding Pythagoras scarcely
take us out of the realm of legend. The name of Theano is famous; she
is usually called Pythagoras’ wife, but sometimes his daughter, or only
his pupil;?® but Theano is also the name of the wife of the mythical
king Metapontus!?® Bro(n)tinus, named in the book of Alcmacon, is
sometimes called father and sometimes husband of Theano,®° and both
are placed sometimes in Croton and sometimes in Metapontum.”
Pythagoras’ son and successor is most often Telauges,” but he also has
other sons, daughters, and successors.33 Of the other ““ Ancient Pythag-
oreans” collected in the Fragmente der Vorsokratiker, Cercops 15
probably a figment of ancient philology,*4 Petron perhaps an inven-
tion,?> Paron a misinterpretation of Aristotle,9* and Parmiscus is only
attested in the cult legend of Delos.*
26 Oldfather, RE XI 81ff; K. Lange, Gétter Griechenlands (Berlin, 1940) pl. I and ἢ.
117; E. Simon, Charites: Studien zur Altertumswissenschaft (Bonn, 1957) 41f;S. ΡΟ Noe. The
Coinage of Caulonia (New York, 1958); L. Lacroix, Rev. Belge de numism. 105 (1959) 5-24.
27 Below, ch. II 3, n. 150.
28 Cf. von Fritz, ΚΕΝ A 1379-1381; διεβοήθη τοὔνομα, Dicaearchus ap. Por. VP το.
Wife of Pythagoras: Hermesianax fr. 2, 85 D.; D.L. 8.42 = Schol. Pl. Rep. 600b; Suda
s.v. Pythagoras; lam. VP 146, 265. Daughter of Pythagoras: Anon. Phot. 438b31. Pupil,
wife of Brotinus: D.L. 8.42; Sudas.v. Theano; Iam. VP 267 p. 146.22. Those who ascribed
complete celibacy to Pythagoras naturally could not make Theano either his wife or his
daughter.
22 RE V A 1379. On Metapontus, cf. Kroll, RE XV 1326.
30 DK 17 (Brontinus); Alemaeon fr. 1; father of Theano, D.L. 8.42; husband of Theano,
ibid. Suda s.v. Theano, Iam. VP 267.
31 Above, n. 17.
32 Von Fritz, RE V A 194-196; Dittmar, Aischines 214ff; successor of Pythagoras:
Anon. Phot. 438b30; son of Theano: Schol. Pl. Rep. 600b = Suda s.v. Pythagoras
(Hesychius); D.L. 8.43; cf. Iam. VP 146, Euseb. Praep. evang. 10.14.14, Por. VP 4, ete.
33 His son Mnesarchus: Anon. Phot. 438b29, Iam. VP 265, Schol. Pl Rep. 600b =
Suda s.v. Pythagoras, Euseb. Praep. evang. 10.14.14; his son Arimmnestus: Duris FGrHist
76F23 (Por. VP 3); his son Mamercus: Plut. Aem. Paul 1; “Damon” Schol. Pl. Rep.
6oob {a corruption of “Damo,” his daughter [D.L. 8.42 τ΄ letter of Lysis, Hercher,
Lpistologr. er. p. 603, lam. VP 146]); other daughters: Aesara, Arignote, Myia: cf. Well-
mann, RE s.v. Aisara, Arignote, Damo; von Fritz, RE τιν. Myia 2.--lam. VP 265:
διάδοχος... πρὸς πάντων ὁμολογεῖται... γεγονέναι ᾿Αρισταῖος.
4 (ch. 11 3, τ. 60,
DK 16; ef. Jacoby on PCrHist 654¥s (Hippys of Rhegium, contra, Schmid 12,
Fort); ΝΜ. Vollrath, Mremosyne 4.2 [raga] out; Burkert, Philologus tos, 18st).
WOH. W424,
ὑτ (1, Wa, a tye.
2. Historical Background
The tradition of the political activity of Pythagoras is consistently
related to Croton, rather than to Metapontum. When it is reported
that Pythagoras made a series of speeches before the boys, the young
men, the women, and the magistrates,3® this may well reflect an
archaic, club-like organization of society.°® But the reconstruction of
the political events is difficult.
Two events were basically important in determining the history of
Croton during this period, its defeat by Locri at the Sagras River*®
and the victory over Sybaris which is dated 510 B.c. From that time
on, according to the evidence of coins,“1 Croton exercised some kind
of hegemony in southern Italy. It clearly ended, suddenly, about 450
s.¢., shortly before the revival of Sybaris-Thurii. The Pythagoras
tradition contains reports, with various small modifications, of a great
catastrophe: the house of Milo, which was the meeting place of the
ythagoreans in Croton, was burnt down by their opponents, and
only a few of those present escaped. Since one of the survivors, Lysis,
‘4 Dicaearchus, fr. 33 (Por. VP 18), names magistrates, youths, boys, and women; the
speeches are given in a different order (youths, magistrates, children, women) in Iambli-
chus (VP 37-57, from Apollonius, after Timaeus: above, ch. II 1, nn. 12, 37). In an ex-
planation by the Socratic philosopher Antisthenes of the word πολύτροπος in Od. 1.1
(scholium on the line from the Homeric Questions of Porphyry), is found the sentence
οὕτω καὶ Πυθαγόρας λέγεται πρὸς παῖδας ἀξιωθεὶς ποιήσασθαι λόγους διαθεῖναι πρὸς
αὐτοὺς λόγους παιδικούς, καὶ πρὸς γυναῖκας γυναιξὶν ἁρμοδίους, καὶ πρὸς ἄρχοντας
upyovtixous, καὶ πρὸς ἐφήβους ἐφηβικούς. There is no cogent reason to strike this sentence
of the Antisthenes material, as being an addition by Porphyry (with H. Schrader; L.
Ikadermacher, Artium scriptores, SBWien 227 [1951] 121f). Cf. also F. D. Caizzi, Antisthenis
fragmenta (Milan, 1966) p. 107. Interest of some Socratic in Pythagoras may be reflected
in Plut. De curios. §16c. But the content of Pythagoras’ teaching was unknown, according
tw |icaearchus ap. Por. VP το. The speeches given in extenso by Iamblichus (VP 37-57)
are a later production (pace Rostagni, ScrMin 1 3ff, De Vogel, Pythagoras 70-147; cf.
(symuasium 74 [1967] 459).
Ὁ Cf. also Morrison, CQ 1956, 145ff. According to Timaeus FGrHist 566F131, Pythag-
oras’ daughter was as virgin the leader of virgins, as matron the leader of the matrons of
(‘roton. Cf. Commentary on Alcman τὸ Page, fr. 1 iii.
4 This event is usually dated ca. $30 B.c., but Kiechle (sof) would put it ca. 550, von
Fritz (RE 183) ca. $20.
41 Kahrstedt, Hermes 1918; von Fritz, Pol. 68ff; C. M. Kraay, “The Coinage of Sybaris
after 510 B.c.” Num. Chron. 18 (1958) 13-37; De Vogel, Pythagoras 55-57.
12 The principal source is Aristoxenus, fr. 18. According to Dicaearchus, who refers
to oral tradition current in his own day, such uprisings took place “‘everywhere’”’ (fr. 34),
and Polybius (2.39) speaks of the burning of Pythagorean συνέδρια, using the plural.
Presumably Aristoxenus wanted to gloss over the extent of the anti-Pythagorean activities
(von Fritz, Pol. 30). Cylon is named as the leader of the opposition; it is noteworthy that
Aristotle names him as opponent of Pythagoras in his list of literary feuds (fr.7§ =» ΓΙ,
2.46, 8.49; in the former passage the reading Κύλων καὶ ᾽Ονάτας is defended by Diels,
IDK 14.18, against Menage’s conjecture Κύλων Kporavuirns; he refers to Tam. VP p.
143.23 Deubner).
II. PYTHAGORAS IN THE EARLIEST TRADITION
later became the teacher of Epaminondas in Thebes,** this termination
of Pythagorean dominance cannot have taken place before 450 B.c.,
and therefore certainly not during the lifetime of Pythagoras. It is
tempting to interpret these events along with the numismatic evidence,
to the effect that from about 510 to 450 B.c., under the leadership of a
Pythagorean oligarchy, Croton experienced a period of prosperity
and power, which then was broken up by internal strife. After this
catastrophe, Tarentum became the center of Pythagoreanism.
It is a question, however, what Pythagoras himself had to do with
these events. After the defeat at the Sagras, according to Justin’s
account, he succeeded in reforming the Crotoniates’ character to such
a degree, by “daily praises of virtue,’ that the victory over Sybaris
became possible. Then, according to Timaeus, directly after this
victory, they fell into luxurious living (τρυφή). According to a
detailed account in Diodorus, Pythagoras, functioning as the embodied
conscience of Croton, had a part in bringing about the war against
Sybaris. A number of factors conspire, however, to make this account
seem very suspicious: the quite different account of Herodotus, the
inherent improbabilities, and, most of all, the structure of Diodorus’
exposition, which is very like the plot of a tragedy.*® The destruction
of Sybaris was the worst atrocity wrought by Greeks against a Greek
city in that era; the attempt to make the unheard-of comprehensible
43 Aristox. fr. 18 (lam. VP 249f), D.L. 8.39, etc. (DK 46). Alcidamas, as reported in
Arist. Rhet. 1398b18f, must refer to this (Θήβησιν ἅμα οἱ προστάται φιλόσοφοι ἐγένοντο Kat
εὐδαιμόνησεν ἡ πόλις; cf. Diod. 15.52). Frank, who spoke of ‘‘literary fiction,” has
neglected to take account of the earliness of the evidence (294.1). Wuilleumier (564f)
expresses doubts of the story’s historicity. On Philolaus’ residence in Thebes, cf. Pl.
Phd. 61e, on Simmias and Cebes, above, ch. I 4, n. 39.
44 Justin 20.4.5ff, Timaeus FGrHist 566F44-45. According to Apollonius (Iam. VP 255)
the revolt against the Pythagoreans resulted from conflict over conquered Sybarite
territory. Dunbabin (360) points out that no Olympic victors from Croton are listed
for the years §48-532. This can be combined with the report of Justin and so used for
dating the battle at the Sagras (above, n. 40). But is it certain that Croton’s recovery was
due to Pythagoras?
45 Diodorus 12.9.2ff. The source is neither Ephorus (Diels, hesitantly, DK 14.14) nor
Timaeus (Lévy, Sources of), but some later author (E. Schwartz, RE V 685f). Herodotus’
version (5.44f) is different. An illogical feature of Diodorus’ account is that the tyrant
Telys of Sybaris (Herodotus calls him king) first banishes his opponents and then, after
they have sought refuge at Croton, demands their return. What follows is modeled,
down to details, on tragedies like Aeschylus’ Suppliants and Euripides’ Heraclidae and
Suppliants. Vhe exiles have taken refuge at the altars in the marketplace of Croton,
enissaries of Telys demand their surrender and threaten war, the popular assembly is in
doubt what to do until Pythagoras addresses them and reminds them of the obligation to
protect suppliants, and the war and the fimous victory follow. According to Andron
(ch. ΠΠ 3 below), Pythagoras foretold to a tricnd the conquest of Sybaris, which makes him
ascer, without active participation in dhe events, as in tua PP 133,177 (Apollonius?)
2. Historical Background
was bound to give rise to legends, and the contrast of ‘‘Sybaritic”’
luxury with Pythagorean sobriety was a strong stimulus to the
creation of moralistic and edifying fiction.
According to Aristotle and Aristoxenus, Pythagoras withdrew to
Metapontum before the outbreak of trouble and died there;4* but
according to Dicaearchus, Pythagoras himself was in Croton at the
time of the revolt,4? and in several accounts he is even said to have
perished in the fire.*® Since the escape of Lysis cannot have occurred
before 450 or so, the reports of Dicaearchus and the later writers cannot
be accepted. Probably two anti-Pythagorean movements have been
combined, one in Pythagoras’ lifetime and another about 450.4 It is
clear, however, that there was no established tradition, no author
scems to use documentary evidence;>° everything depends on oral
tradition, in which “Pythagoras” quickly comes to mean ‘“‘the
Pythagoreans,” in much the same way as the Ionians speak of Ion
and the Dorians of Heracles.
When Plato contrasts Pythagoras with the lawgivers as one who
became, ἐδίᾳ τισὶν, a guide to the good life, he is not thinking of him
as a political figure.*! In some respects, Pythagoras is connected with
46 Arist. fr. 191, Aristox. fr. 18; similarly Apollonius (lam. VP 255), Philod. Rhet. II
180 Sudhaus, Justin 20.4. Others accounted for Pythagoras’ absence from Croton by
saying he had gone to Delos to care for, and bury, his old teacher Pherecydes (Satyrus
ap. D.L. 8.40, Nicom. ap. Por. VP 55, lam. VP 252). This was contradicted as early as
IDicaearchus (fr. 34 = Por. VP 56; to judge by the form of the citation, in indirect speech,
the sentence in question belongs with the Dicaearchus quotation, and not, as Lévy thought,
Sources 51, n. 4, to Porphyry).
47 Frr, 34-35. Others expanded the story: his disciples made a bridge over the fire
with their own bodies, so that the master might escape (Por. VP 57; Tzetzes Chil.
11.80ff). Hermippus’ version is unique: Pythagoras died in the war between Acragas and
Syracuse, because he refused to escape by crossing a field of beans. (D.L. 8.40, Schol. PI.
Rep. 600b, Suda s.v. Pythagoras; at D.L. 8.93 this is combined with his flight from the
fire. It is hardly possible to determine whether the similar story of Myllias and Timycha
at the beanfield, recounted by Neanthes FGrHist 84F31 = Iam. VP 189ff, is primary,
as Lévy thought, Lég. 68, or derivative.)
48 T).L. 8.39, Schol. Pl. Rep. 600b, Suda s.v. Pythagoras (Neanthes? cf. above, ch. II 1,
n. 6), Hippol. Ref. 1.2.16, Plut. Stoic. rep. 1051c; cf. Arnob. 1.40, Mara bar Serapion
(W. Cureton, Spicilegium Syriacam [London, 1855] p. 736).
49 Minar $3ff; with modifications Morrison, CQ 1956, 149.
50 Dicaearchus (fr. 34) refers expressly to oral tradition. The so-called ὑπομνήματα of
the Crotoniates cited by Apollonius (lam. VP 262) are suspect. Reference to such a source
may point to Timaeus (cf. Morrison, CQ 1956, 149); but the whole thing may be a
fabrication, for the ‘memoirs’ are supposed to prove that the arbitration judgment by
the representatives of Tarentum, Metapontum, and Caulonia, which was unfavorable to
the Crotoniates, was purchased by bribery; cf Delatte, Pol. 218; von Fritz (Pol. 65f)
thinks the “memoirs” “may have been contained ina local chronicle which might have
been embellished by legends.” Cf. Lévy, Sources 115.9.
Rep. dooa, This is cmphasized by Delatte, Pol, 17f, Prank, AJP 1943, 222, τ. 7;
R. Joly, Mém, Ac. R. de Belgique $1.4 (1048) 28.
I. PYTHAGORAS IN THE EARLIEST TRADITION
Metapontum more closely than with Croton,®* while the political
activity and the burning-episode belong to Croton.5* Some modern
scholars have gone so far in a skeptical direction as to assert that
Pythagorean political activity in Croton is an invention—Aristoxenus
and Dicaearchus, they suppose, projected upon the Pythagoreans
their own ideal of the Bios πρακτικός, and thus invented
Pythagorean politics together with an appropriate historical back-
ground.*4
There is, however, an often forgotten testimony of Theopompus,
in the midst of a fragment of Posidonius about the tyrant Athenion
of Athens.® Athenion had been a member of the Peripatetic school,
but at the first opportunity he cast aside the mask of philosophy and
became a tyrant, thus illustrating “the Pythagorean doctrine regarding
treachery, and the meaning of that philosophic system which the noble
Pythagoras introduced, as recorded by Theopompus in the eighth book
of his History of Philip, and by Hermippus the disciple of Callimachus.”*
Theopompus must have said that the secret, but genuine, goal of the
philosophy introduced by “the excellent Pythagoras” (ὁ καλὸς
Πυθαγόρας) was tyranny.
There are other traces of an anti-Pythagorean tradition, representing
Pythagoras and his pupils to be tyrants. Appian writes, in a context
very similar to that of Posidonius, ‘also in Italy, some of the Pythag-
oreans, and in other parts of the Grecian world some of those known
as the Seven Wise Men, who undertook to manage public affairs,
governed more cruelly, and made themselves greater tyrants than
ordinary despots.”’®? Diogenes Laertius names a Πυθαγόρας Κροτωνιάτης,
τυραννικὸς ἄνθρωπος, who is said to have been a contemporary of the
philosopher—a desperate attempt to get rid of the tradition of Pythag-
62 Pythagoras is once called a Mctapontine (Lycus [?] ap. Por. VP 5; cf. below, ch. Il 5,
ἢ. 66), but never a Crotoniate. He died in Metapontum (above, ἢ. 18). Theano, too,
belongs to Metapontum (above, nn. 28-29). .
683 Considering the novelistic quality of his whole account, there is no significance in
Plutarch’s localization of the burning-cpisode in Metapontuin (De gen. $83a).
4 CF ch. 1, nn. 43, 44, 98 The skepticism was expressed most extremely by Frank,
AJP 1943, 2226.
πὴ Theopompus PGrHist 115673, cited in Posidonius ΠΌΑ 8716. 0 Ath. 5.2136,
given here in Gulick’s Loeb tr. Cf Delatte, Vie 2st. Hermippus, who is intermediary
‘ , - 3 + ‘
source between Theopompus and Posidonius, wrote βίοι τῶν ἀπὸ φιλοσοφίας εἰς τυραννίδας
καὶ δυναστείας μεθεστηκότων (Acad. Ind. col. t1, p. 29 Mekler).
' : = κι ‘ tgs ͵
M On the history, see U. von Wilamowitz-Mocllendorff, ' Athenion und Aristion,
ΡΠ 1927, 10. $0. . .
δ᾽ Mithr. 28, in White's Loch tro He is speaking of the tysimny of the Mpicurean
Aristion, who came to power shortly after Athenion, ‘The source may well, again, be
Hermippus,
2. Historical Background
oras as tyrant;>® and clsewhere the revolt against the Pythagoreans is
represented as a blow for freedom from tyranny.®9
The existence of this tradition as early as the fourth century helps us
understand why it was so important to Aristoxenus to show Pythag-
oras as an opponent of Polycrates and an émigré in search of freedom,
and to emphasize the readiness of the Greeks of southern Italy to
accept Pythagorean dominance or leadership.®°
‘lheopompus’ interpretation is as tendentious as that of Aristoxenus.*!
But, if the Pythagorean dominance in Croton can be seen in two such
different lights in the fourth century, as a model of free government
under aristocratic guidance or as a detestable tyranny, the underlying
reality cannot be an invention of Aristoxenus and Dicaearchus, nor
one by Theopompus, but only a tradition that could be interpreted in
more than one way. We must believe Dicaearchus’ testimony that the
memory of Pythagorean rule and of revolt against it was still alive in
Magna Graecia in his day. Plato, too, alludes to the Pythagoreans’
connection of politics and philosophy (above, ch. I 4).
‘There is no inconsistency between this and the religious and ritual
side of Pythagoreanism. In fact, cult society and political club are in
origin virtually identical. Every organized group expresses itself in
terms of a common worship, and every cult society is active politically
sa évacpia.© Pythagoreanism fits into this picture and can be seen to
have firm rootage in the social and political conditions of the time.
1).L. 8.46; cf. Delatte, Vie 251. For similar concern about Pythagoras’ “image,”
wud the assertion that he authorized athletes to eat meat, cf. below, ch. II 4, ἢ. 111.
™ 1).L. 8.39, Tert. Apol. 46.13, Arnob. 1.40; cf. Delatte, Vie 251, Pol. τό. Apollonius,
tao, hints at this theme (lam. VP 257ff).
Tree choice as the basis of Pythagorean rule: Jam. VP 249 = Aristox. fr. 18. Von Fritz
(Hol. 186), discussing Aristoxenus’ bias, suggests that the whole passage from Nicomachus
(Vor, VP 2if= Iam. VP 33f) in which Pythagoras is represented as converting
various tyrants and frecing many cities was drawn from Aristoxenus; and in this he is
lollowed by Wehrli (ad Aristox. fr. 17). But considering the manner of citation, this
wcemms uncertain: the sentence for which Aristoxenus is specifically cited, about Lucanians,
Meucetians, Messapians, and Romans as pupils of Pythagoras, is missing in the parallel
paussape, lam. VP 33f but stands by itself at D.L. 8.14 and Iam. VP 241. It is, then, an
wolited citation, of interest to later generations because of the mention of the Romans;
note also that it breaks the continuity at Por. VP 21f. It is hard to attribute to Aristoxenus
the anachronism of including Tauromentum (above, ch. ἢ 1, ἢ. 37; Frank makes much
ot this, AJP 1943, 220).
ΤΙΝ thrust at Pychagoras may be intended as a covert blow at the Academy, against
which Theopompus wrote a pamphlet (κατὰ τῆς Π]λάτωνος διατριβῆς, Ath. 11.508c—d),
and against which the reproach was so often directed, that Plato’s pupils became tyrants
@f. Ath. loc. cit., from Herodicus).
"Thus in Athens the desecration of the Perms and the profanation of the mysteries
constituted a political act, as che “ Bacchanalia’” were regarded in Rome as a coniuratio
(νυ 49.8.1, 14.3). Plato’s Academy was a cult organization (on this, Boyancé, Muses
249), and Dion's partisans were united, among other things, by initiation into the mys-
tenies (PL ip. 7.333¢, Plut. Dion $4),
II. PYTHAGORAS IN THE EARLIEST TRADITION
In some reports the name of Pythagoras has an almost mythic
function: a belief, a political act, or a mishap experienced by Pythag-
oreans is described by an assertion that Pythagoras taught, did, or
suffered this or that. But we must resist the temptation to interpret
Pythagoras as a mythical figure—the “speaker from Pytho” as an
incarnation of the Hyperborean Apollo,® his fiery death as deifica-
tion, and the like. Even what is legendary has its relation to specific
historical events. There is no doubt of the historical reality of the
Pythagorean society and its political activity in Croton; but the
Master himself can be discerned, primarily, not by the clear light of
history but in the misty twilight between religious veneration and the
distorting light of hostile polemic. Pythagoras and the Pythagoras
legend cannot be separated.
3. METEMPSYCHOSIS AND “‘SHAMANISM
That Pythagoras taught the doctrine of metempsychosis is generally
regarded, and rightly, as the one most certain fact in the history of
early Pythagoreanism. For it is alluded to in the most ancient piece of
evidence about Pythagoras, the well-known lines of Xenophanes,}
, 7 /
Kai ποτέ μιν στυφελιζομένου σκύλακος παριόντα
ἜΝ κ“ \ “ὃ , θ ” .
φασὶν ἐποικτῖραι καὶ τόδε φάσθαι ἔπος
- ν as , 3 7 3 \
παῦσαι μηδὲ ῥάπιζ᾽, ἐπεὶ ἦ φίλου ἀνέρος ἐστίν
? 4 # , >?
ψυχή, THY ἔγνων φθεγξαμένης ἀίων.
63 This interpretation appears as early as Aristippus the younger (D.L. 8.21 = Giannan-
toni A162): ὅτι τὴν ἀλήθειαν ἠγόρευεν ody ἧττον τοῦ Πυθίου. Apollonius has it somewhat
differently, in the context of the birth legends (lam. VP 7: ὅτι... ὑπὸ τοῦ Πυθίου
προηγορεύθη). Pythagoras “sounds like a nom de guerre,” says Nilsson (A History of
Greek Religion [Oxford, 1925, 1950] 202). But there are many personal names of similar
form (Athenagoras, Diagoras, Hermagoras), and the name Pythagoras itself is not rare
(cf. the RE articles). |
64 Cf. Heracles on Octa, whom Pereginus Proteus imitated in A.D. 167, as Timanthes
of Cleonae had done (Paus. 6.8.4). Lévy, Lég. 71.1, refers to Croesus, who was transported
from the pyre, according to the legend, to the land of the Hyperborecans, R. Eisler
(Orpheus the Fisher [London, 1021} 11f) rashly equates Pythagoras with Apollo, and Apollo
in turn with the sun-god, who makes his way to the west and is there burnt in his house. . .
ΕΚ 21B7- DL. 8.36 (A. P.7.120; Sudas.vv. Xenophanes, στυφελίξαι). Whether
Pythagoras is the person referred to was questioned by O. Kern, AGP 1 (1888) 499, by
Rathmann, 37f, and by Maddalena, 335 Maddalena adds that, even if Xenophanes did
mean Pythagoras, the doctrine in question was not metempsychosis but only the equality
of the human and the animal soul; bat this overlooks the explicit formulation φίλου
ἀνέρος ψυχή ἐστιν. The passage is regarded. as the best piece ofevidence about Pythaporas by
Zeller (1 ς47), Wilamowits (GHEE IL tgo), Kranz (Hermes 1944, 220. DK 1 4yo.38ff),
Mondolfo (in ZM 21.441). ‘Thomas (71f), Long (17), Dodds (irr. 143. $9), and others.
3. Metempsychosis and “ Shamanism”
Iiopenes Lacrtius says these lines refer to Pythagoras. In the frag-
ment itself there is no name mentioned, but Diogenes explicitly cites
the beginning of the poem, so that at least his source, where Xeno-
planes is cited for evidence on Pythagoras, was based on the complete
text. The first words (“And once ...”) show that other, similar
anecdotes had preceded this one; thus the subject was a well-known
person, and not some anonymous “Orphic.”’ It is worth noting that
Xcnophanes also attacked Epimenides;? not only the Homeric religion
was subject to his criticism, but non-Homeric religious manifestations
such as doctrines about the soul and about ritual purification.
An allusion of Aristotle is equally explicit. He complains of his pre-
decessors that in their theories about the soul they paid far too little
attention to the necessary presuppositions about the body: “They try
to say what kind of thing the soul is, but do not go on to specify about
the body which is to receive the soul, as though it were possible, as
in the tales of the Pythagoreans, for just any soul to clothe itself in just
any body.’’3 In his critique of various philosophers, he introduces this
ironical comparison with the Pythagoreans’ “‘myths’’ as though they
were something well known. What he cares about here is not distinc-
tions among individuals, but among species. The soul “clothes itself”
(a common expression in the doctrine of metempsychosis)§ in any
kind of body, human or animal; and this failure to distinguish between
luuman and animal gives rise to the same scandalized tone that we hear
in Xcnophanes. Theophrastus, on the other hand, in arguing against
sacrificing. animals, tries to bring animals and human beings closer
toyether. The former, too, have souls, quite like those of human
Ajwillonius of Tyana imitated his Master, recognizing in a lion a later incarnation of
King Anisis (Philostr. VA 5.42; see Lévy, Sources 2 n. 7, 134).
"U1. “18 (DK 21 1}, 1.111 (DK 21B20).
" Atist. De ait. 407620 (DK $8B39) and 414422. Doubts that the passage refers to metem-
psychosis were expressed by . .athmann 18f, and Maddalena 338ff. The latter claims that
tuetenpsychosis implies a juc.zment of the dead, so that it is not the entrance of “just
ay soulinto “just any”? body; but what Aristotle is talking about is the character of
wily an peneral, and not of those of individuals. Maddalena suggests that he is thinking
mt the cluse particles in the air (ξύσματα), alluded to at De an. 404a18, though this doctrine
ie tather compatible than otherwise with metempsychosis. Cherniss (Pres. 325 n. 130)
haves the matter undecided. See also Nilsson, Op. I 663, and Theiler, Arist. 98, and the
Moyliias anecdote, Arist. fr. ror.
μῦθος is used by Aristotle in a sense very similar to the modern one. Cf. 1074438.
"Kranz, DK 1 $04.7; εἰσδύεσθαι Hdt. 2.123; similarly D.L. 8.77, Por. Abst. 1.1, Max.
byt toa, Diod, 5.28.6. Democritus can also speak of atoms as ἐνδυόμενα εἰς τοὺς πόρους
(Aros), ch Hippoc. Piet, 1.25; but the ancient commentators have no hesitation about the
reference of the present passage τὸ Lictempsychosis, Philop, De an. 140.34 cites Empedocles
117 an explanation of it; ef Olympiod. In Phd. A Xt, p. «6.178 Norvin.
II. PYTHAGORAS IN THE EARLIEST TRADITION
beings, or perhaps even identical with them, “as Pythagoras taught.’”¢
Porphyry gives a description, taken from Dicaearchus, of the arrival
of Pythagoras in Croton: Pythagoras had great success, and was
invited to give lectures before the civic leaders, the young men, the
boys, and the women.
As a result of these events, a great reputation grew up about him,
and he won many disciples from the city itself; not only men but
women too... [including Theano], as well as many from the non-
Greek territory nearby, kings and nobles. Now the content of his
teaching to his associates no one can describe realiably, for the
secrecy [σιωπή] they maintained was quite exceptional. But the
doctrines that became best known to the public were, first, that the
soul is immortal, then that it migrates into other species of animals,
in addition that at certain intervals what has once happened happens
again, so that nothing is really new, and finally that we ought to
regard all living things as akin. Pythagoras is said to have been the
first to introduce these opinions into Greece.
Whether all this is from Dicaearchus is controversial and cannot be
definitely decided by philological means.’ It is in Porphyry’s manner,
however, to quote long passages; and his second citation from Dicaear-
chus is an extensive section, as can luckily be proven from the parallels.®
The skeptical tone of the expression is noteworthy; no one, he says,
knows for sure. This cannot come from Porphyry, who cites, from
different sources, details about the mathematical knowledge and pro-
cedures of Pythagoras;® nor from a source like the “handbook” uscd
6 Por, Abst. 3.26: εἶ φαίνοιτο κατὰ Πυθαγόραν καὶ τὴν ψυχὴν τὴν αὐτὴν εἰληχότα {τὰ
ζῷα), from the concluding sentence of the argument attributed at 3.25 to Theophrastus.
The sentence is hardly an addition by Porphyry, who expressly denied the extension of
metempsychosis to animals (De regressu an. fr. 11 Bidez).
7 Por. VP 19; section 18 is given as fr. 33 by Wehrli. The transition from indirect
to direct discourse, which may seem suspicious, has already taken place in 18. Section
19 was attributed to Dicaearchus by Rohde, Q 126, Jager 40, Burnet, EGP 92, Rostagni,
ScrMin I 42 n.1, Lévy, Sources §0, Mondolfo in ΖΜ 363f. The attribution was contested
by Rathmann 3f, on the ground that the report is incompatible with Dicacarchus'
partisanship for the vita activa (though we know at least, from fr. 36, that Dicacarchus did
treat of Pythagoras’ metempsychosis doctrine). Maddalena (76 n. 21) suggests that the
uncertainty expressed does not comport with the public instruction reported in section
18 (but this is intended as ironical), and also that Aristotle had some exact knowledge of
Pythagorean doctrines (but not, we may reply, doctrines of Pythagoras himself). Wehrli,
following Rathmiann, omits section 19.
* Por, VP 566057 th. a4 Wor the conclusion 1},1. δ fr. 35 W.
"Por, VP ot, citing the Memoirs. Inthe next sentence Fudoxus is cited (ir. 30 Gisinger,
426 Lasserve); Gisinger is wrong in assigning the whole passage, including the citation of
the Memoirs, to Hudoxus.
3. Metempsychosis and “ Shamanism”
hy Diogencs Laertius, which knows of writings by Pythagoras.’ In
(act, such an expression is scarcely conceivable after the image of
Pythagoras that originated with Speusippus, Xenocrates, and Herac-
lides had become canonical. This points, then, to the antiquity of the
source, and specifically to Aristotle’s pupil Dicaearchus, who was so
skeptical about the Platonic-Pythagorean doctrine of the soul." One
of the doctrines designated here as “best known to the public,” that
af the cyclic recurrence of all things, is attested as Pythagorean by
I cacarchus’ colleague, Eudemus.”
lon of Chios had already named Pythagoras as the one who knew all
about the soul, in an epigram which, according to Diogenes Laertius,
refers to Pherecydes of Syros:1%
a € A ? ͵ , 3 \ ‘ 3 -
my Oo μεν ὨΡΟΡΕΊ) TE KEKQO [LEVOS ἠδὲ Kat αἰδοῖ
‘ / ~ 4 ” fa
καὶ φθίμενος ψυχῇ τερπνὸν ἔχει βίοτον,
uv / 3 4 ’ [3] \ /
εἴπερ Πυθαγόρης ἐτύμως σοφός, Os περὶ πάντων
ἀνθρώπων γνώμας εἶδε καὶ ἐξέμαθεν....
Likewise, the tradition of the Greeks that lived on the Black Sea and
the I{cllespont, as reported by Herodotus,* connects the Γέται
huvariZovres with Pythagoras. In neither of these cases is anything
wid about metempsychosis, but Pythagoras is obviously the best-
known name in the realm of immortality and the afterlife.
lt is no less important to recognize, however, that the later tradition
lrequently attempts to ignore or interpret away the doctrine. There
in not a word about metempsychosis in Aétius; and, what is more, he
projects the Platonic doctrine of the soul onto Pythagoras. The
ML, 8.6f => Schol. Pl. Rep. 600b = Suda s.v. Pythagoras.
"ot ch. Ht, m. 48. A similar agnostic expression in Por. VP ς is probably also old:
“Vt you happen not to know the country or the city whose citizen this man was, do not
lie camecrned; for some say he was from Samos, some from Phlius, some from Metapon-
tain.” ‘The name of the author of this is, however, uncertain: MSS “εῦκος (AvxKos?
Atincav?). See below, ch. II 5, ἢ. 66.
he KRW) DK 58834 -- Simpl. Phys. 732.26); cf. fr. 89 ΝΜ. (D.L. 1.9).
M YOK 36184 fr. 30 Blumenthal =. fr. 5 Dichl (D.L. 1.120). Rathmann disputes the
inference to Pherecydes (44f), but this cannot be checked. He says the last verse is an
itetpalation, because the μέν of line 1 has no corresponding δέ; but for a fragment this
hid of atgument is inconclusive. Maddalena (348f) emphasizes that metempsychosis is
not explicitly mentioned.--In line 3 the manuscript reading is εἴπερ Πυθαγόρης ἐτύμως
Hnopos περὶ πάντων; the obviously correct cmendation is that of Sandbach, Proc. of the
tome. Milol, Soc. 185 ns. § (195841959) 36. The wording of line 4 is awkward. For the
anbyect matter Kranz (Hermes 1934, 227} compares Eleraclitus fr. 129 (followed by
ΝΜ ἢ in ZAM, 3178, and Sandbach, loc. cit). A similar expression about Pythagoras
be πη} at (᾽ν, Uist. 4.3.02,
MWogas go; below, mn. 203.
II. PYTHAGORAS IN THE EARLIEST TRADITION
Memoirs have no place for palingenesis; souls that are pure rise to the
“highest,” and the impure are given over to the Erinyes.!® A Stoic
source in Sextus!® gives as the reason for Pythagorean vegetarianism
the basic relationship of all living beings, from the gods down to
animals without reason, by virtue of the πνεῦμα τὸ διὰ παντὸς τοῦ
κόομου διῆκον. Xenocrates had adduced the relationship of man and
beast as a reason not to eat meat, and Theophrastus expressed a similar
idea.” As against Empedocles’ eloquent advocacy of vegetarianism,
with the gruesome picture of a father killing his own son (B136-137),
these later philosophers have sublimated metempsychosis into a
generalized, philosophical doctrine of soul.!* Xenocrates, to be sure,
developed a theory of daimones, but did not assume metempsychosis;!°
and, whereas the main line of the tradition makes Pythagoras a
reincarnation of Euphorbus, Xenocrates makes him the son of Apollo.?°
Once more the same conclusion: insofar as the later tradition contra-
dicts the most ancient evidence, it agrees with the thought of the
Platonists, and especially Xenocrates; he reinterpreted the Pythagorean
tradition to make it conform to his way of thinking. The fact that in
this case the later tradition is not unanimous in discarding metempsy-
chosis, along with Xenocrates, is not due to a continuous Pythagorean
tradition, but to the tremendous influence of the Platonic myths and
the dissenting views of other members of the Academy and Peripatos,
especially Heraclides Ponticus.”!
15 31; see above, ch. I 3. In the Memoirs, as in Aétius, only the highest part of the soul
is immortal. The neo-Pythagorean Alexander of Abonuteichus rejected metempsychosis
in somewhat similar terms: the ‘‘soul’”’ develops and then perishes, but the important
thing is the “spirit” (φρήν) which emanates from the mind of “Zeus.” (Lucian Alex.
40; in the background is Pl. Tim. 30b; cf. Plut. De fac. 944e.)
16 Sext. Emp. Math. 9.127f; on this preuma theory which was attributed to Pythagoras
see Cic. Nat. d. 1.27.
17 Fr, 98 H. = Por. Abst. 4.22. Xenocrates also speaks of the danger of “assimilation”
to the souls of irrational creatures through the eating of meat (fr. 100 H. = Clem. Al.
Strom. 7.32.9; here Polemo is named as well as Xenocrates). For Theophrastus, see above,
n. 6.
18 Sext. Emp. Math. 9.127, 129 cites these same lines of Empedocles, but docs not
mention metempsychosis.
19 On the theory of daimoues, see Heinze 78ff; for the rejection of metempsychosis,
147.
20 Fr. 22 H. = lam. VP 7 (cf. Por. VP 2). Eudoxus and Epimenides are also cited for
this view; cf. the statement of Speusippus about the birth of Plato, fr. 27 1νν 0 D.L. 3.2.
Cf. below, ἢ. 143.
21 Clearchus, like Eleraclides, championed the immortality of the soul and its indepen-
dence of the body (Welrli p. 47). In fr. 38( 0 Ath. 4.1§7¢, DK 441514) he represents the
Pythagorcan Euxitheus doubtless a fictitious person as speaking about the imprison-
ment of the soul in the body and the prohibition of suicide (cf PL. Phd. ὁ 16),
3. Metempsychosis and ‘“ Shamanism”
‘The doctrine of metempsychosis is set forth, with some detail, in
works of Pindar,?? Empedocles,?* Herodotus (2.123), and Plato.*4 The
uestion how much of this can be accepted as testimony on Pythag-
uteanism depends on one’s judgment of the complicated and much
iliscussed phenomenon of “‘Orphism.” Scholars’ conceptions of
l'ythayoreanism and of Orphism are inevitably as interdependent as
the pans of a balance. A “‘ minimalist” attitude to the Orphic tradition
tapully raises the importance of Pythagoreanism,”® while hyper-
ciiticism toward Pythagoreanism peoples Greece with Orpheotelestae.?®
here is no such thing as a communis opinio on Orphism, especially since
the sensational discovery of the papyrus of Derveni?’ has shaken many
eatablished views.?°
A tew details, however, are likely to survive examination. There
were 'Opdixd, purported poems of Orpheus, perhaps circulating in
litlering versions, and including at least a theogony and cosmogony,
af the sort for which the papyrus of Derveni provides a philosophical
ceunmentary.2® There were ’Op¢eoreAcorai,2° who with reference to
these writings gave private initiations to mysteries, in which the
punishments in store for the uninitiated in the next world were vividly
depicted. For the initiates there was the Bios ᾿᾽Ορφικός, an ascetic life
featuring specific abstinences, and especially vegetarianism.
of 2, ΘΗ, fr. 133 Schr. (= 127 Bowra). Cf. Long 29ff, Nilsson I 692f, E. Thummer,
Phe Religiositét Pindars (Innsbruck, 1957) 121-130.
" Katharmoi, passim. Cf. Long 458.
“Phd. 8ib, Menex. 81a, Rep. 614ff, Phdr. 248d, Gorg. 525¢ (with Dodds’s note). Cf.
Jhecias passim, Long 63ff, von Fritz, Phronesis 1957; also R. S. Bluck, “The Phaedrus
awl Meamearnation,” AJP 79 (1958) 156-164, “Plato, Pindar, and Metempsychosis,”’
hich qos 414.
4 Walamowitz (GidH II 182ff) is followed in this trend especially by Thomas, Linforth,
awd! Loup, As a result, not only metempsychosis, but even the Gold Tablets (ch. II 2,
oft) tum Pythagorean.
“hee esp. Rathmann.
4A parte of it is published by 5. G. Kapsomenos, Arch. delt. 19 (1964) 17-25. Cf.
μ Methelbach, Zeitschr. f. Papyrologie u. Epigraphik 1 (1967) 17-32; W. Burkert,
tf 1 1yOR, 93-114. :
[he most critical study of the Orphic tradition is Linforth’s, followed by L. Moulinier,
thphée et POrphisme ἃ Pépoque classique (Paris, 1955). The most favorable treatment of the
ἐν tradition, since Kern, is that of Ziegler, RE XVIII 1321ff. A middle position is
ΜΙ by Guthrie, Orpheus; Nilsson, Op. Π 628ff; Dodds, Irr. 147ff. For bibliography,
ae Nilwon, Op, 11628, GrR 1 678ff.
Cdphic writings are mentioned in Eur. Alc. 967, Hipp. 954, Pl. Rep. 364¢ (βίβλων
δὲ jeuddy παρέχονται Μουσαίου καὶ 'Opdéws), Arist. De an. 410628 (ἐν τοῖς ᾿Ορφικοῖς
ἔνε καλουμένωι) cf Gen. an. 734019, and fr. 7. See Nilsson, Op. I 630fF.
Phe word is found in Theophr. Char. 16.01, Plut. Lac. apophth. 224. For the facts,
we PL Rep. 40a (μὴ θύσαντας δὲ δεινὰ περιμένει),
"᾿ξ )ρμφικουὶ βίοι, Pl Leg. 782¢; ἄψυχος βορά Kur. Hipp, 9§2.
Π. PYTHAGORAS IN THE EARLIEST TRADITION
Metempsychosis is not attested directly for Orphism in any ancient
source?2—only the preexistence of the soul. It is undergoing punish-
ment in its confinement to the body, which is both a prison and a
protection.33 Souls are borne by the wind into the body;* it is an
almost unavoidable supplement to suppose that other living creatures,
at their death, have “breathed οὐ these same souls. According to
Plato, priests of the mysteries teach reincarnation: “priests and
priestesses who have paid careful attention to being able to explain
their ministry’’ (Meno 81a).
Herodotus ascribed to the Egyptians a fully developed theory of
metempsychosis; and, since the Egyptians never had such a doctrine,
it is clear that Greek ideas are being projected onto the foreigners. His
manner of introducing the doctrine suggests south Italy.” Herodotus
makes the ominous addition: “There have been some Greeks, both
earlier and later, who have subscribed to this doctrine, giving it out
as their own. Though I know their names, I am not putting them
down.” All possible combinations have been tried in the attempt to
decide what Herodotus is concealing, but with no certain result.38
32 Wilamowitz, GldH II 194: “eine orphische Seelenlehre soll erst einer nachweisen.”
That Orphism taught metempsychosis is contested by Kriiger, Orph. 37f, Thomas 9 9,
Long 89ff. On the other side of the argument are Rohde, Psyche Hf 121ff (Eng. ed. 341ff),
Nilsson, Op. II 663ff, GrR I 691ff, Dodds, Irr. 149 with ἢ. 94.
33 Plato Crat. 400 b—c; also Phd. 62b, Xenocrates fr. 20. Neither the Attic φρουρά nor
περίβολος can have occurred in a hexameter poem.
34 Arist. De an. 410b27; cf. Kern, Orph. frag. 223.
35 Teg. 870d-e; cf. Wilamowitz, Platon I 249. Perhaps we may add the verse of Rhada-
manthys, Arist. EN 1132b27.
86 Cf. H. Bonnet, in Reallexikon der dgyptischen Religionsgeschichte (Berlin, 1952) 76f.
This is regarded as a closed question; Kees, for example, does not even mention it.
37 2.123: “The Egyptians say that Demeter and Dionysus are the rulers of the under-
world. And also, they were the first to proclaim the following doctrine...” Herodotus
adds the doctrine of metempsychosis, because the preceding context reminded him of
Greek ideas. Demeter and Dionysus in close connection point to southern Italy; it is
from there that the triad of Ceres, Liber, and Libera was introduced to Rome in 493 B.c.
38 Available for choice are the Orphics, Pythagoras, and Empedocles. (Pherecydes is
hardly eligible; cf. Long 13f.) The possible combinations are (1) Orphics and Pythagoras
(Zeller, SBBIi 1889, 993; Nilsson, I 701; Morrison, CQ 19$6, 137; and others), (2) the
Orphics and Empedocles(Rathmann 48ff; though he does not exclude the first possibility),
(3) Pythagoras and Empedocles (Long 22, Timpanaro Cardini 21f). Maddalena (346f)
concludes that Pythagoras is not intended, from the fact that Herodotus does name him
at 4.05. It has been thought that it was Empedocles whom Herodotus did not wish to
namie, for the reason that he was still alive (Seein in his note on the passage; Burnet,
EGP 88 νι. 4). Herodotus is reticent in the realn of cultic ἀπόρρητα (c.g. 2.61, 2.47, 2.0 70f),
but also as to what he considers inferior (4.43, 1.613 thus Long, 23 n. 60, thinks [lerodotus
does not wish to mention a plagiarist; Morrison, CQ 1056, 137, suspects he is think ing
of Eleusis). Hone adopts the longer version at 2.80, Orphismn “is” Pythagorean for Hero-
dotus, so that Pythagoras and Empedocles must be intended; and the echo of a theory of
the elements Gi journey through creatures of and, water, and air) does remind one of
Einpedocles tr 11s, even though there are details that do not mitch exactly (ao mention
of fire; 7,000 instead Of yO,000 years 100 years Kern, Orphy frag. 240).
3. Metempsychosis and “ Shamanism”
li) another passage Herodotus mentions Pythagorean and Orphic
teaching in the same breath, and compares it with Egyptian,** but the
decisive sentence is transmitted in two versions, and it is matter for
Viperous controversy whether one branch shows interpolation or the
ather abbreviation. The Egyptians may not enter any sanctuary in
woolen clothing, nor bury their dead in woolen shrouds, he says.
Then the Florentine branch continues, ὁμολογέουσι δὲ ταῦτα τοῖσι
"Ομφικοῖσι καλεομένοισι καὶ Πυθαγορείοισι, but the Roman branch has
ἠμυλυγέει δὲ ταῦτα τοῖσι ᾿Ορφικοῖσι καλεομένοισι καὶ Βακχικοῖσι, ἐοῦσι
δὲ “ἸΠἰγυπτίοισι καὶ Πυθαγορείοισι. One cannot, on principle, prefer one
nf the manuscript families.t° It should be clear, though, that a con-
tamination of the two versions, as found in the textus receptus, is
unacceptable.“
The short version is, naturally, smoother,* but the more compli-
eated thought of the longer version also makes sense: “these [customs]
agree with those [Greek customs] that are called “Orphic’ and
'Wacchic,’ but are in reality Egyptian and Pythagorean.” In this version
rufira is the subject, and the datives are therefore to be taken as neuter;
whereas in the other case masculine datives correspond to the personal
ailjcet (the Egyptians).4? This comparison favors the longer version,
fin the ancient testimonia speak of ᾽Ορφικά, not ᾽Ορφικοί. 4. The follow-
inp, words, too, τούτων τῶν ὀργίων, would come more naturally after
a preceding neuter; in addition, the use of ὁμολογέειν seems more
## y 41. Wilamowitz believes in interpolation (GidH II 189 n. 1 = 187? n. 1), as do
μ᾿ πη} s2f and PhW $4 (1934) 1178ff, Linforth 38ff, Long 24f, Timpanaro Cardini
444 The longer text is defended by Kriiger, Orph. 13f, Boyancé, Muses 94.1, Dodds, Irr.
yn Ko, Guthrie (Orpheus τό, 272 n. 4) accepts the contaminated textus receptus with-
ΒΗ} qestion,
ἢ Schinid 1 2.672 and Jacoby, RE Supp. II 515ff. According to the evidence of the
payrytt, the ¢wofold tradition docs not go back to ancient times (A. H. Paap, De Herodoti
“ἢν in papyris cf membranis Aegyptiis servatis [Leiden, 1948] 95ff), and in Byzantine
Hines there was no longer any reason for interpolation. There is no way to tell whether,
in lis tact reference to the passage (Apel. 56), Apuleius was using the shorter text (as
Vuteoth 47) or was himself abridging.
That is, the longer version but with the verb ὁμολογέουσι. After Zeller SBBin 18809,
wag, abd Burnet, HGP 88 n. 4, had separated the words cai Πυθαγορείοισι by a comma,
Pe jtaul moditicd to καὶ <rotow> Πυθαγορείοισι (Coll. Budé, 1936). Maddalena, 236f,
ralnily omits as interpolated everything after ᾿Ορφικοῖσι.
“More reasonable and intelligible,” says Linforth (45); but this is merely the result
Hb Wa brevity.
Tadorth would like τὸ deny this (44); but sce 2.80: συμφέρονται... Αἰγύπτιοι...
Nanedaiporioat, “βλλήνων οὐδαμοῖσι. ‘The expression καλεόμενοι, ἐόντες δέ is not
shewhere directly attested, but ἐόντες, καλεύμενοι δέ is not infrequent (2.178, 3.99).
“ Wilanowite GEE tag.
Dodds, fr. τον an. Ro: Rathmann, Phil $4 (ty34) 1084, artificially separates τούτων
rear ἐρμγίων: “the ritualy of these.”
II. PYTHAGORAS IN THE EARLIEST TRADITION
natural in the longer text.46 On the other hand, it is hard to reach a
definite conclusion as to the extent to which the variants ὁμολογέει and
ὁμολογέουσι fit in better with the assumption of interpolation or
omission.*”
In any case Herodotus states that there is a connection between
Orphism and Pythagoreanism in the realm of ritual. In addition, the
longer text contributes not only an indication that Orphism is con-
nected with Dionysus, but a theory about its origin, namely that it
comes from Pythagoras, who got his teachings from Egypt. There is
of course a relation between this and the presentation of the doctrine
of metempsychosis, and probably also other features such as the state-
ment that the Egyptian priests abhor beans, which has not been
confirmed by Egyptian evidence.4* Thus Greeks in Egypt connected
beliefs about the afterlife and religious customs with the name of
Pythagoras in the same way as Greeks living in the area about the
Hellespont and the Black Sea did with the beliefs of the Getae about
immortality (Hdt. 4.95). The difference is that in the latter case the
Greeks’ feeling of superiority to the barbarians led them to make
Zalmoxis the pupil of Pythagoras, while in Egypt their awe of the
ancient foreign culture produced an opposite result, and Pythagoras
became the pupil of the Egyptians.
In general, modern scholars have seen Orphism as the older and
more comprehensive movement and considered Pythagoras to have
been influenced by it,4® but the reverse theory is not only in the longer
46 Cf. Plut. De Is. et Os. 364f: ὁμολογεῖ δὲ καὶ τὰ Τιτανικὰ καὶ Νυκτέλια τοῖς λεγομένοις
. and Plut. Caes. 9 (on the Bona Dea festival) αἱ γυναῖκες πολλὰ τοῖς ᾿Ορφικοῖς
ὁμολογοῦντα δρᾶν λέγονται. Rathmann, PhW 54 (1934) 1182f, emphasizes that ὁμολογέειν
is elsewhere used by Herodotus only of persons; but in the sense ‘‘agree’”” = “‘resemble,”’
the accusative of specification ταῦτα would be odd, too. The verb is construed with οὐδέν
and xara at 1.142, 2.18, 6.54.
47 Linforth argues acutely (46) that an interpolator would also have had to alter ὁμολογέουσι
to ὁμολογέει (a point missed by all those who accept the textus receptus), whereas an
abridgment would not have needed to alter ὁμολογέει to ὁμολογέουσι, so that the dis-
crepancy has been caused by interpolation. Yet we read, in the passage beginning in
section 80, συμφέρονται δὲ καὶ Tobe... τόδε... συμφέρονται, so that it was easy to
alter a following ὁμολογέει δὲ ταῦτα, by assimilation, to ὁμολογέουσι, and when the
neuters became masculines the abbreviation was necessary.
48 Hdt. 2.37, with B. A. van Groningen’s notes (Leiden, 1946).
49 A “Ιερὸς λόγος which Iamblichus read told of Pythagoras being initiated by the
Orphic Aglaophamus (lam. VP 146). It was regarded as certain that Pythagoras got his
doctrine of metempsychosis from Orphism by Rohde (Psyche I 107F — 336f, Eng. ed.),
Zeller 1 68M, 563, Rathmann, Guthrie (Orpheus 216ff), Nilsson [ 7orf, Kern (Die Religion
der Griechen I | Berlin, 1935] 162). See also Jaeger Theol, τὼς. r4sf Eng. ed.; HTR 42
(1989) 135-147; B.S. Bluck, Plato's Meno (Cambridge, 1961) 61ff, 2746. On the other
3. Metempsychosis and “ Shamanism”
version of Herodotus; Ion of Chios also made use of it. In his Τριαγμοί,
whose authenticity is guaranteed by Isocrates,®° he said that Pythagoras
“lhad written some poems and attributed them to Orpheus” (ἔνια
niujuavra ἀνενεγκεῖν εἰς "Opdéa).5! This report was unearthed by
ancient scholars interested in the controversy over whether Pythagoras
lelt any writings; but it is a misunderstanding to conclude from it that
lon knew works of Pythagoras, who referred in them to Orpheus.™
Ti εἰς "Ophea (or Μουσαῖον) ἀναφερόμενα is a common designation for
the literature which circulated in the name of Orpheus or Musaeus.*?
What lon meant was that the real author of certain poems circulating
tmcler the name of Orpheus was Pythagoras—that the ᾽Ορφικά were
HvWaydpeca. It was not works of Pythagoras that Ion knew, but Orphic
poems whose origin he was trying to determine. Herodotus says
emphatically that the poets who were reputed to have lived before
Itomer were actually later (2.53); and this is aimed specifically at
Orpheus. Clearly, to an educated person of the age of the Sophists it
already seemed incredible that works had been preserved that were
aide, Lobeck maintained the thesis that Orphism was the borrower from Pythagoras
(Mel; alo, 2478, 330f, 358f, 698). Also see R. Turcan, RHR 150 (1956) 137.
ΜΕΤ, 15.268 (DK 36A6). Harpocration (s.v. Ion, DK36A1) writes of the work
Triaumnos: ὅπερ Καλλίμαχος ἀντιλέγεσθαί φησιν ὡς ’Envyévous. Epigenes wrote περὶ τῆς
εἰν 'Opdda ἀναφερομένης ποιήσεως, giving an allegorical interpretation (Clem. Al. Strom.
1141.4, §.49), and also wrote commentary for Ion’s tragedies (Ath. 11.468c). Linforth 116ff
identities him, conjecturally, with Epigenes the pupil of Socrates (Pl. Ap. 33e, Phd.
sul, Xen. Mem. 3.12.1. Dodds’s objection, Irr. 171, is refuted by the Derveni papyrus).
Hi lhe is tight, the text of Harpocration may be kept, though the usual thing is to emend
it (iad ᾿᾿᾿πιγένους Bergk, ὡς καὶ ᾿Ἐπιγένης Diels, καὶ ᾿Επιγένης Jacoby; Pfeiffer, in his
tute on Callin. fr. 449, sides with Diels and Jacoby). It is worth noting that Clement,
ter (Strom, 1,131.5), brings together Ion (DK 36Bz2) and Epigenes (DK I 105.31). The
wenirnce in the Suda ἔγραψε (sc. Orpheus) Τριαγμούς, λέγονται δὲ εἶναι “Iwvos τοῦ τραγικοῦ
(εν (Mpheus [DK 1Ar]) can hardly be anything but a misunderstanding of the item in
Hatpocration (Lobeck 388f, Kern, Orph. frag. p. 318).
EK 1612 - fr. 24 Blumenthal = D.L. 8.8; almost the same wording in Clem. Al.
Anum 1,131.4 > FGrHist 392F25 = Kern, Orph. frag. test, 248 (Iwv... Πυθαγόραν
εἰς ‘Opbda ἀνενεγκεῖν τινα ἱστορεῖ). Cf. Linforth 1o9ff-
** Pannery (MSc EX 226ff) combined this with the story of Aglaophamus (lam. VP
64h) and ascribed both to the same forger. Gigon says (Ursprung 125), ‘Ion is acquaintd
with writings of Pythagoras, and claims to know from them that Pythagoras attributed
the τ πον they expound to the ancient bard Orpheus”; and Kranz expresses himself
ἀπ] terms (Hermes 1934, 227). The correct interpretation is given by Lobeck (389),
ΕΠ ἢ. (t88), and Nilsson (f 701).
ἀπ ὴ εἰς ‘Opdda καὶ Μουσαῖον ἀναφερόμενα Phid. De pict. p. 80 G. = Orph. frag. 30;
Vpipenes’ title, above, ἢ. 40, Tatian 4 Orpht. frag. test. 183; cf. Phid. De pict. p. 66
Vi IK 04.07, DK at Br, Clem. Al. Strom. 1.131, Schol. BT If, 18.570. Sudas.v. Orpheus
IK 1At: ἀναφέρονται εἰς ᾽Ονομάκριτον; here, too, ἀναφέρειν εἴς τινα means “attribute
authorship to someone.”
I. PYTHAGORAS IN THE EARLIEST TRADITION
written by a member of the Argonautic expedition. On the other
hand, they had to account for the existence of the “Orphic”’ litera-
ture,55 and there were various ways to do this. One could construct a
number of characters all named Orpheus, and this was done as early
as the fifth century by Herodorus of Heraclia.** Or, one could put the
responsibility upon other writers—ancient epic poets,>? Onomacritus,”*
or Pythagoras and various Pythagoreans. These methods also crossed
over one another, and this is probably the explanation of the reports
about an “Orpheus of Croton,’”’®® or “Cercops the Pythagorean.’ 6°
Epigenes, especially, was prone to explain everything by Pythagorean-
ism:*! and Plato, too, shows that he knows Magna Graecia 15 a suitable
place for such material to come from, when he attributes a myth
about the underworld to a κομψὸς ἀνήρ, tows Σικελός τις ἢ ᾿Ιταλικός.53
Perhaps Heraclitus, who in one fragment (40) reproaches Pythagoras
along with Hesiod for useless πολυμαθίη, had already connected
Pythagoras with Orphism: Πυθαγόρης Μνησάρχου ἱστορίην ἤσκησεν
μάλιστα πάντων καὶ ἐκλεξάμενος ταύτας τὰς συγγραφὰς ἐποιήσατο ἑαυτοῦ
σοφίην, πολυμαθίην, κακοτεχνίην. The genuineness of the fragment has
been suspected, because its purpose in Diogenes Laertius 1s to prove
54 See also Arist. fr. 7.
55 Unlike the Pythagoreans, the Orphics committed their teachings to writing from
the beginning, as is shown, aside from the evidence cited in n. 29 above, by the vase
paintings showing a scribe standing before Orpheus’ singing head, Linforth 122ff. (In-
cidentally, their use of writing gives us a historical terminus post quem.)
56 FGrHist 31F42.
3? Prodicus, the composer of a Minyas, in which a trip to Hades was described (Kinkel,
EGF p. 215ff), becomes the composer of the Orphic katabasis (Suda s.v. Orpheus = DK
1A1; Wilamowitz, GldH II 197 n. 3). On Cercops see n. 60 below.
58 On the role of Onomacritus see Nilsson, Op. II 645ff. It is certain that he collected
and edited; but the idea that he composed whole poems (as Philop. De an. 186.26, Suda
s.v. Orpheus) or ‘“‘invented”’ the story of Zagreus (as Paus. 8.37.5), is philological conjec-
ture.
89 Asclepiades ap. Suda s.v. Orpheus, where we also find an Orpheus from Camarina.
Nilsson assumes (Op. II 644) that the poems in question actually came from Croton and
Camarina.
60 The testimonia are in DK τς. A tradition transmitted by Aristotle names a Cercops
as a rival of Hesiod (Arist. fr. 75 = D.L. 2.46), and the epic poem Aegimius was sometimes
ascribed to Hesiod (frr. 294-301 M.~W.) and sometimes to Cercops of Miletus (Hes.
fr. 301 M-W. = Ath. 11.503d; cf. also Schmid I 1.254). It was natural to identify the
rival of Hesiod with the composer of certain poems that stand in rivalry to Hesiod’s
Theogony—the Orphic theogonies; and, if then the "Opdixd were classified as Pythagorean,
Cercops became straightway a Pythagorean. There is no need to distinguish two men
named Cercops (pace DK I 106.6ff, Timpanaro Cardini 69 n.).
“l Epigenes ap. Clem, Al. Strom. 1,131.5 attributes the κατάβασις and the “]ερὸς Λόγον
to Cercops, the Πέπλος and the Φυσικά to Brotinus (IK 17).
ἋΣ (org. 4914: cf. below, ch, HT 2, αι. 48.
3. Metempsychosis and “ Shamanism”
that Pythagoras left writings.®? But since it does not prove this at all,
lecause all it claims is that Pythagoras plundered the writings of others,
and not that he wrote anything himself, its authenticity has been
yencrally recognized since the discussions of Wilamowitz and Rein-
lardet.* There remains what seemed to Wilamowitz the “excruciating
dithiculty”” (“peinliche Schwierigkeit”’) of guessing what “writings”
Heraclitus could have been thinking of. Surely not Homer and
Hesiod, who were known to all Greeks. But aside from them, and
Anaximander, there was scarcely anything in the purview of Greek
culture that Pythagoras could have used for such a purpose, except
'Cpdud and the like. Perhaps the word Heraclitus chose, κακοτεχνίη,
points in the direction of the ἀγύρται καὶ pdvres.©
If this is right, Heraclitus saw the relation of Pythagoras and Orphism
in the opposite way from Herodotus and Ion.® Where they attribute
Orphic doctrine to Pythagoras, he makes Pythagoras the borrower.
‘The contradiction is not particularly surprising, for even at that early
late no one could be precise about chronology, in matters of this kind.
(ach arranged the facts in the way that best suited his purpose: whoever
wanted to find a tangible personality in the chaotic mass of ’Op¢uxd
lit upon Pythagoras, and those who wanted to cast doubt on his
ΟΝ μία ΠΥ used Orphism for this purpose.
‘Thus the oldest sources show Pythagoras, unlike Orpheus, as a
lanpible personality of the historical period, but their doctrines as
connected or even identical. There is no support in these sources for
the modern attempts to discern a difference in doctrine between
“ Wizy 1}. 8.6; it was branded spurious by Diels, AGP 1890, 451, and included
anne the “dubious or forged” in DK. Zeller (SBBin 1889, 988; I 393 n. 5) wished to
strike: out at least the mention of writings: ἐκλεξάμενος ταῦτα[ς τὰς cvyypadas |.
"Vor its genuineness, after Bywater: Burnet, EGP 134 n. 2; Reinhardt, Parm. 235.1;
Wilsmowitz, GldH II 188 n. 1; Rathmann 39; Kranz, Hermes 1934, 227f; Morrison,
(1) 196, 136. Reinhardt called the reference to ἱστορίην in ταύτας τὰς συγγραφάς
“are λιν Πρ PEirte” (cf. fr. 5). There could have been something in the preceding context
that would have made the word συγγραφαί easier to understand. (H. Gomperz, Hermes
aM Jtust] qi, trics to show that fragments 40, 41, 129, and 81 followed each other, without
a laeuk, an that order; but this cannot be proven.)
See below, ἡ. 226. Onomacritus worked ἐκλεγόμενος, according to Hdt. 7.6 (a
retevence supplied me by M, West). The reference is to Orpheus, according to Rathmann,
ud, per contra, Kranz, Hermes 1934, 116, and Mondolfo in ΖΜ 317 point out that (accord-
jig to later tradition) Pythagoras collected Egyptian, Babylonian, and Persian traditions.
Ate we to suppose, however, that Heraclitus thought Pythagoras capable of doing re-
ated in foreign-language books? (To be sure, a certain Antiphon states that Pythagoras
learned Epypeian, D.L. 8.3.)
“Rathmann 410.
II. PYTHAGORAS IN THE EARLIEST TRADITION
Orphism and early Pythagoreanism.*” It is only too easy for modern
notions to intrude. If one believes, with Nietzsche, in a primal opposi-
tion of “ Apollonian” and “‘ Dionysian,” then Pythagoras and Orphism
must stand in the same polar relationship; and if, under the influence
of later evidence, one regards the philosophy of number and the
foundation of exact science as the essential ingredient of Pythagorean-
ism, the antithesis of Apollonian rationality and Dionysian mysticism
fits in very nicely. We must bear in mind, however, that as the Greeks
thought of them, Apollo and Dionysus were brothers; the supposed
clear differentiation of Pythagoreanism from Orphism is simply not
attested in the oldest sources.
In the historical and social realm, much more than in that of doctrine,
we do seem to detect a certain difference between the two groups.
Much of the evidence about the Orphics, as in general about the
unofficial mystery cults, clearly reflects the activities of mendicant
priests.*° Pythagoreans, however, both in Croton and probably else-
where in southern Italy, held position of dominance in their cities, for
a time. This means that something related to Orphism had emerged
from the anonymity of back-alley ritual and become respectable;
known persons, not apocryphal writings, are active in this movement—
Πυθαγόρειοι, not Opducd. Legend knew of royal seers like Melampus;
the abnormal is sometimes rated higher and sometimes lower than the
normal. While the cults based on the supposed writings of the mythical
singer of antiquity remained suspect in the eyes of many, in the person
of Pythagoras the ancient figure of the wandering purveyor of
67 “A pollinizzare l’orfismo” is Ciaceri’s characterization of the achievement of Pythag-
oras (“‘Orfismo e Pitagorismo nei loro rapporti politico-sociali,” Atti R. Acc. di Arch.,
Lett. e Belle Artin.s. 12 [Naples, 1931-1932] 209-223). At the same time, Ciaceri interprets
Orphism as “democratic” and Pythagoreanism as “aristocratic.” Linforth (43) considers
the conjunction of “Bacchic” and “Pythagorean” in the longer version of Hde. 2.81
to be an argument for athetesis. Maddalena (363 n. 98, 327 n. 25) would use athetesis in
this same passage to such effect as to remove completely any connection of Orphic and
Pythagorean, and Delatte also (Litt. 4f) is for carefully separating them. Cameron (486)
assumes an independent development from a common source; and both Rohde (Q 104;
somewhat differently in Psyche II 107 = 336f. Eng. ed.) and Dodds (Irr. 143, 149 n. 9$)
also interpret the two movements as parallel developments. Frank (357 n. 168) and
Boyancé (Muses 93ff; cf. REG 1941, 161 π. 2) stated that they were inextricably inter-
twined. Nock emphasizes that in contrast to Orpheus, Pythagoras is “a tangible figure 7
(“Herodotus 2.81,” Studies Presented to Ἐν L. Griffith |[London, 1932] 248). Note the
confident assertion in Herodotus 4.96 that Pythagoras lived many years later than Zalmox.
88 Cf the anecdote of King Leotychidas and the Orpheotclestes Philippus, Plut. Lae.
apophth. 224¢. It need not be truc historically, but is good evidence for the general atmno=
sphere. For the typical figure of the ἀγύρτην, which is in itself quite old, cf. Aesch. Ag.
1273f, Ar. Pax 1ogstf, Av. osoff, Demosth. 18.260, 19.249, 19.281 (on Aeschines’ mother),
Antiphanes fr. 1sy, Clearchus fr. 47 ΝΜ.
3. Metempsychosis and “ Shamanism”
salvation had taken on a new radiance which cast its gleam as far as
l'lato’s philosopher-king.®*
A general observation may advance our argument somewhat.
Mystery cults offering the promise of salvation, even along with
cusmogonic myths, are conceivable without a doctrine of metem-
psychosis; and the fact that Orpheus, but not transmigration, is often
tonnected with Eleusis?® might be an indication that there was at least
one branch of “Orphism” without metempsychosis. In that case, this
doctrine must be an innovation upon a general Orphic background;
lur its introduction, Pythagoras is the only obvious candidate. There
lau possibility, deserving serious consideration, that it came from India.”
lt is the Pythagorean variant of Orphism that is manifest in Pindar,
lunpedocles,”? Herodotus, and Plato, connected always with south
Italian or Sicilian tradition. What details are to be attributed to
Pythagoras himself remains an open question.”* Does every living
creature have an immortal soul that migrates from one incarnation to
another? Do plants have such souls?** Do they only enter certain
apecics of animals?”> Do only certain special individuals, even among
inet (δαίμονες, Empedocles calls them) undergo this wondrous experi-
δι ὁ} To what extent does the doctrine imply clearly formulated
01, ch. 1 4. On the aristocratic character of Pythagoreanism see Minar, Ciaceri
(alive, nn. 67), and Kerényi 35. G. Thomson, in arbitrary manner, calls Pythagoras
“the titst great exponent of democratic thought” (Aeschylus and Athens, 2nd ed. [London,
0. 46} 410}.
Ai, Ran. 1032; Dem. 25.11; Marmor Parium, FGrHist 239A14-15; Hecataeus of
Alaleta, Diod. 1.96, cf. 4.25; Plut. fr. 212 Sandbach.
1) Cornelius, cited by K. von Fritz, Gnomon 40 (1968) 8 n. 1; Kranz, Emped. 2of.
lhulda (rr. 172 n. 97) suggests that Greek and Indian doctrines of metempsychosis have
4 henidnon origin in shamanism.
© Jmpedocles is regularly connected in the ancient tradition with Pythagoras, from
Alviduinas on (below, ch. II 2, n. 7) but only indirectly with Orpheus: Apollonius has
UN pheus sing about Neikos (1.498), and fr. 141 is also cited as a verse of Orpheus (Geop.
444). The modern tendency to speak of Orphism in relation to Empedocles (see O. Kern,
“Upedokles und die Orphiker,” AGP 1 [1888] 498-508) comes from the fact that the
lnsapy oot Pythagoreanism is determined by ideas about number philosophy and exact science.
See C. Hopf, Antike Seelenwanderungsvorstellungen, Diss. Leipzig, 1934; W. Stettner,
Lie Seelenwanderung bei Griechen und Roémern (Stuttgart, 1934); Long, passim; von Fritz,
Phones 1057; B.S. Bluck, AJP 79 (1958) 405-414.
“ According to ordinary usage, the word ζῷα (Hdt. 2.123, Dicaearchus ap. Por. VP
ty), as well as ἔμψυχα, would exclude plants (Rohde, Psyche II 180 n. 4 = 404 n. 82 Eng.
bil), but Empedocles (τὶ 117, 127) and Heractides (fr. 89 W.) include them (Long 23),
ail perhaps this is implied in the taboo on beans (below, ch. II 4).
tam, VP ἃς (there are no human souls in sacrificial animals).
Kerényi (18) showed that the 'Eastern-Western” concept of the similarity of all
athe in by no means selfevident; Rathmann (10) had emphasized that the story of
Haphotbus does not imply a comprehensive doctrine of metempsychosis. On Arist. fr.
ive, see below, n. 129.
ll. PYTHAGORAS IN THE EARLIEST TRADITION
beliefs, and how important is the word puy7??? Was there present
at its beginning the significant semantic innovation whereby the
“soul,” as distinguished from the body and independently of it, is
regarded as the “complete coalescence of life-soul and consciousness "78
—a world away from the Homeric conception—or is “soul” primarily
a mysterious, meta-empirical Self, independent of consciousness, as
some important witnesses seem to indicate?’ Is the soul newly
incarnated immediately after the death of the old body,®° in which case
Hades becomes unnecessary, or is there an intermediate phase, which
would leave Hades there, as a way station?®! Is the process of palin-
genesis the work of blind natural forces—a creature at the moment of
birth sucks in the soul®’—or is it the execution of a penalty assessed in
a judgment of the dead? Is there an endless, cyclical movement,* or
is there a fall at the beginning and a salvation at the end which is per-
manent—or perhaps has as its alternative an eternal damnation, in
which case the concepts of Elysium and Tartarus again become
77 Empedocles speaks of δαίμονες. In Xenophanes fr. 7 it is not stated that the dog “has ᾿
a soul; it “‘is” the soul of the friend, and “cries out” (Frankel DPh? 311).
78 Jaeger Theol. 99ff = Eng. ed. 83. See Long 2ff. Itis beyond doubt that metempsychosis,
or at any rate related religious concepts, played a significant role in the development of
the notion of ψυχή (see Dodds, Irr. 140ff; on ψυχή in Heraclitus, B. Snell, Entdeckung des
Geistes [Hamburg, 1955°] 36ff = 17ff Eng. ed.), but they do not presuppose it.
79 Only exceptional persons like Pythagoras remember their previous incarnations.
Pindar even sees a polar relationship between the soul and “consciousness”: The αἰῶνος
εἴδωλον, which comes from the gods, sleeps when one is awake, and is active while he
sleeps (fr. 131; Aesch. Eum. 104; Hippoc. Insomn. VI 640 L.; Arist. fr. 10). In relation to
Indian ideas of metempsychosis, too, one finds it necessary to speak of a meta~empirical
“self? (Long Io).
80 This is suggested in Hdt. 2.123 and Schol. BT Il. 16.857. Empedocles has no under~
world: for him earthly existence is Hades (as Rohde, Psyche II 178f = 381 Eng. ed.;
Dodds, Irr. 174 n. 114; wrongly rejected by Long soff).
81 Cf, Serv. Aen. 3.68. Some of the acusmata speak of Tartarus (Arist. An. post. 94633),
and of dead persons under the earth (Ael. VH 4.17); Pindar and Plato insert a stay in
Hades between reincarnations. The katabasis stories involving an “underground dwelling”
are rooted in the traditional conception of Hades. The bean taboo may also be relevant
here (ch. II 4).
82 Hdt. 2.123: ἐς ἄλλο ζῷον ἀεὶ γινόμενον ἐσδύεται. Schol. T Il. 16.857: Πυθαγόρας
φησίν, ὡς ἀναχωροῦσα ἡ ψυχὴ ἐν ἐκείνῳ γίνεται [ἐν] ᾧ ἂν γεννωμένῳ σώματι ἢ φυτῷ
καταντήσῃ (similarly Schol. B); the soul of Patroclus complains of its fate, because it
fears an unworthy rebirth. Cf. also Arist. De au. 407b22 (above, ἢ. 3). Stettner regards
this amoral metempsychosis as the primary type (7ff, 20ff). Kern, Orph. frag. 223 tries to
compromise.
88 So Pindar, Plato, and the acisma about breaking bread (Arist. fr. 195; lam. VP 86).
Arist. EN 1132barff connects Pythagoreanism with a saying of Rhadamanthys (above,
nn. 35).
"4 ΤΊ is suggested in Ht. 2.123, Budemus fr. 88, Dicacarchus ap, Por, VP 19. Cf the
astrologers in Varro (Aug, De εἰν, D. 22.28, Serv. Aen, 3.08),
4. Metempsychosis and “ Shamanism”
relevant?85 Each of these questions is answered in more than one way
in the ancient tradition, and there would be small prospect of success
in an undertaking to crystallize out a sharply defined “doctrine of
Pythagoras.” There is a veritable maze of conflicting tendencies—
inythical tradition persisting along with newly developing concepts of
the world, the tradition of the mystery cults, ethical demands, and a
yrowing recognition of natural law. It is only too easy for the modern
a holar, from the vantage point of his own rationalistic and systematic
activities, to suppose that at the beginning there was a unified, care-
fully worked out, and firmly defined theory.
(n such a topic as men’s conceptions of a future life, a topic bearing
ini matters so completely outside normal experience, a consistent and
unified doctrine can only exist in the form of scripturally established
ilayima maintained by ecclesiastical authority. In oral tradition, or in
the belief of a group no matter how closely knit, though agreement in
ementials may persist, varying interpretations are bound to emerge.
Conceptions of the afterlife are and have always been syncretistic.*° It
in only theology, coming along rather late in the tradition, that is
interested in smoothing out the differences. Thus it is not to be pre-
supposed that Pythagoras would commit himself to answer all possible
yestions; the main point was not a well-rounded system, but the
working power of the doctrine. If there was no established “Holy
Word,” new interpretations were bound to develop among the
disciples, on the ipse dixit, αὐτὸς ἔφα principle. Only dead dogma is
jreserved without change; doctrine taken seriously is always being
ivvised in the continuous process of reinterpretation.
‘The metempsychosis doctrine does present a new start, after all,
even in the hybrid character that its various aspects reveal. It does not
otler mythical narrative, a picturesque story which gives the inter-
pretation of a ritual, but a gencral doctrine which claims to be
tnnmediately truc.8? While even in Hesiod, no matter how carefully
“© (he version of Pindar and Plato (though Rep. 611a and Tim. 41d introduce the idea
that the number of souls in the world must remain constant). The idea of a fall and a
salvation is most prominent in Empedocles and in Plato’s Phacdrus. Reward and punish-
wentan the next world are very important in Orphism (above nn. 30 and 68).
“Ly Egypt, even the oldest pyramid texts show a good many mutually inconsistent
ideas bout the afterlife (see Kees).
"This ix teuc even if at first the metempsychosis doctrine did not apply to all living
lcips, or even to all men. The story of Euphorbus was intended not as an exceptional
cae but ay an example, On the other hand, there do seem to be special cases in the
avcounts of Epimenides (below, no 166) and Aesopus (ni. 176). Plato is thinking of
metenparychosis in this way when he evaluates it as an effort λόγον διδόναι (Meno 81a).
II. PYTHAGORAS IN THE EARLIEST TRADITION
each detail of his exposition is planned, the relationships and the laws
of the universe are only indirectly stated, in genealogies and mythical
episodes, the doctrine of metempsychosis directly states a general law.
It does not merely explain particular situations; past and present, pre-
existence and life after death are comprehended in a single thought.
And, insofar as animals and plants are included in it, the unity and
homogeneity of the universe are grasped.®* In the thought of the
preexistence and immortality of the soul lies an attempt at consistent
thinking, a groping for something like an eternal, imperishable Being
—a foreshadowing of the ontology of Parmenides.*® To this extent,
even leaving aside the ideas of science and philosophy, Pythagoras, as
teacher of metempsychosis, is not simply the prophet of old-fashioned
piety, but at the same time, in his own way, a thinker leading, through
ancient forms, to a new level of consciousness.
A doctrine like that of metempsychosis, which transcends normal
human ways of knowing, can find a guarantee only in supernatural
experience, in the world of the divine or quasi-divine. If Pythagoras
knew the facts about the fate of the soul in this life and the next, he
must have had superhuman powers and faculties; the prophet must be
able to refer to his own example.® It is natural, then, that elements of
the Pythagoras legend are, from the beginning, connected with the
doctrine of metempsychosis.
It is often seen as matter for regret that miraculous tales have
attached themselves to the figure of Pythagoras and make it difficult
for the scholar to disentangle the thread of historicity from the web of
legend and fiction.* In the circumstances it is very tempting to use
expressions like “‘neo-Pythagorean” or “late antiquity” to classify
these tales.92 One feels confident, in any case, that they represent a
88 In the saying ascribed to Thales, too, πάντα πλήρη θεῶν, the word πάντα points in this
direction; there are not sacred and profane realms, but a unified world.
89 Cf. Kahn’s judgment of Greek doctrines of metempsychosis (AGP 1960, 34): “by
its rigor and its generality, such a doctrine is no longer primitive.” Nilsson, Op. HI δι
sees the origin of the doctrine of metempsychosis in “pure logic” —an exaggeration, but
it gets at an important aspect of the matter.
80 Cf. Empedocles’ expression (fr. 117) ἤδη yap ποτ᾽ ἐγώ, and also Epimenides, DK 3132.
Orpheus and Musacus are also, of course, sons of gods (Pl. Rep. 363¢, Tim. 40d).
91 Zeller, Vortr. 30; Rathmann 1701 ‘‘fabulis tamquam adopertum.”
8 Zeller (1 365) says that a great part of the miraculous tales is ‘“‘neo-Pythagorean”
(cf. Vortr. 42f, and Schottlaender 342). Kerényi (15) speaks of ‘‘spitantike Pythagoras-
legenden.”” Rathmann (246) tries to show that all the stories of Pythagoras were derived
from the legends of Epimenides, Abaris, Orpheus, and Zalimoxis; but this is refuted by
the “golden thigh" theme. Maddalena (148) sees that everything essential ix attested in
gth-century sources, but will not admit that it may go back further.
3. Metempsychosis and “ Shamanism”?
wcondary growth, layers that must be stripped off until what the
uw sentific historian recognizes as ‘facts’ can be seen. Only ina few cases
lus it been recognized that these miraculous stories do not conceal but
reveal reality, that they give us a clue to the impression made on
contemporaries by an actual person,®? and that they may even contain
facts of a special character. The Pythagoras legend is the oldest available
layer of the tradition on Pythagoras; it is attested earlier than any of
the “historical” details of his life in Aristoxenus and Dicaearchus, and
ls presupposed by the Platonizing reinterpretation of Pythagoras in
the Old Academy.
Ax carly as Empedocles, Pythagoras appears as a superhuman
figure:
ἦν δέ τις ἐν κείνοισιν ἀνὴρ περιώσια εἰδώς,
ὃν δὴ μήκιστον πραπίδων ἐκτήσατο πλοῦτον,
παντοίων τε μάλιστα σοφῶν τ᾽ ἐπιήρανος ἔργων"
ὁππότε γὰρ πάσῃσιν ὀρέξαιτο πραπίδεσσιν,
pet’ ὅ γε τῶν ὄντων πάντων λεύσσεσκεν ἕκαστον
. , f/ 3 3 ?
καί τε δέκ᾽ ἀνθρώπων Kal τ᾽ εἴκοσιν αἰώνεσσιν.
‘Timacus understood the words as applying to Pythagoras; others
thenipht of Parmenides.*%* This clearly shows that Empedocles did not
give any name, but also that the anonymous figure appeared as the
teacher and master of Empedocles. The praise of Epicurus by Lucretius,
aul the praise of Lucretius by Vergil, are in the same tradition; neither
fives a name, any more than Ovid does when he introduces Pythag-
maa ‘Therefore, neither some anonymous Orpheotelestes®? nor a
“ Wolide, in 1871 (Q 105), recognized the “ganz echten Mythencharacter.” See also
Εὐνγ ραν, Gigon, Ursprung 131, Dodds, Irr. 144f, with nn. 63-64. Lévy thought the
Vythaporas saga had, indirectly, even influenced the gospels (a view disputed, rightly, by
be Mo] Lagrange, Rev. bibl. 45 [1936] 481-511; 46 [1937] 5-28, and M. Goguel, RevPh
44 [tueX| 241 270). He tried to show that the historical tradition about Pythagoras, in
Atiatxenus and others, was a rationalization of the legend. Rostagni, too (Verbo 121 ἢ. 1)
Κα ot the “yospel” of Pythagoras.
"Yr, 120 (PLL. 8.54 -- Timacus FGrHist 566F14, and Nicomachus ap. Por. VP
te dar, FP 67).
| * ftathmann 42f, Maddalena 343ff. For Pythagoras as teacher of Empedocles, see
below, cho ΠῚ 3, ἃ. §9; for Parmenides as his teacher (obvious from the fragments)
Plo ἘΝ ap. 1.L.8.§§ Don, 477.18.
Ltr 62th Verg. G. 2.490ff, Ov. Met. τς. 6086 (Rostagni, Verbo 266f, Long 18
Hof) ἀμ} mentions the report chat Pythagoreans avoided pronouncing Pythagoras’
nau {Προ 240; Tam. VP 8&8, probably from Aristotle; also Tam. VP 1g0, 244).
Watheraciny 138. . ᾿
II PYTHAGORAS IN THE EARLIEST TRADI TION
man of the Golden Age®® can be intended. The mental power of the
great man extends over “ten and twenty human lifetimes’ ;°* his
mind is superior to the change of generations—is it his immortal soul
that remembers “every detail’’?
Heraclides has Pythagoras tell this tale:1°° he had once been
Aethalides, the son of Hermes, and received from the latter the gift of
remembering everything, both in life and in death. Thus he knew that,
as Euphorbus, he had been slain by Menelaus in the Trojan War, and
that he had subsequently been Hermotimus, then Pyrrhus, a fisherman
of Delos, and finally Pythagoras. Here we are on the shaky ground of
Academic and Peripatetic controversy. Dicaearchus and Clearchus
give a very different list of Pythagoras’ previous incarnations: Euphor-
bus, Pyrandrus, Aethalides, a beautiful prostitute named Alco, and
Pythagoras It looks as though each one treated the Pythagoras
tradition as his whim or fantasy dictated.
Actually, it can be shown that there was an independent tradition
involving Acthalides,'” Hermotimus, and the Delian fisherman,
98 Zeller, SBBln 1889, 990; I 396 n. 1, $84 π. 5. Against this interpretation Rostagni
(Verbo 229ff) cites fr. 132, which shows that in Empedocles’ view there are at all times
“divine” men. The attribution to Pythagoras has been defended also by Kranz, Hermes
1935, 112 n. 2; Mondolfo in ΖΜ 329: Long 178: Timpanaro Cardini 17f. Maddalena,
too, concedes this (64.3), but contests the connection with metempsychosis, since Nico=
machus interprets the words as referring to Pythagoras’ hearing the harmony of the spheres;
but it may be that Nicomachus, or one of his predecessors, was trying to get rid of the
metempsychosis theory.
99 ἀνθρώπων αἰών = generation ; cf. Aesch. Sept. 744 (Long 21, $1); wrongly “in
seinen zehn und zwanzig Menschenleben,” DK. Homer’s Calchas knows ‘what is, what
will be, what was” (JI. 1.70), the sibyl, too, defies time (Heraclitus fr. 98); cf. also
Parmenides fr. 4.1.
100 Fr, 89 W. = D.L. 8.4; repeated Hippol. Ref. 1.2.11, Por. VP 45 (Euphorbus at the
beginning), Tert. An. 28f (Pyrrhus before Hermotimus), Schol. Soph. El. 62 == Suda
s.v. ἤδη (Πύθιος instead of Πύρρος), Hieron. In Ruf. 3.40 (from Por. VP 45; Callicles
instead of Aethalides), Schol. Ap. Rh. 1.646 (a double version, with Hermotimus once
corrupted to ᾿Ηλεῖός τις and placed after Pyrrhus, and the second time left out). See
Rohde, Psyche II 417ff = so8ff Eng. ed., Rathmann 9 n. 67. Which book of Heraclides is
the source is uncertain (the Abaris according to Corssen, RhM 10912, 28; A. Rehm ibid,
423; Lévy, Sources 40; Wehrli, hesitantly, suggests the ἄπνους).
101 Dicaearchus fr. 36, Clearchus fr. τὸ = Gell. 4.11.14.
102 Pherecydes of Athens, FGrHist 3F109 ᾿ Schol. Ap. Rh. 1.645. (See Jacoby’s note;
the attribution to Pherecydes of Syros, DK 7B8, can scarcely be right.) Aethalides is a
son of Hermes. He has the privilege that his soul may dwell part of the time on earth and
part of the time in Hades; this is reminiscent of the Dioscuri, not of metempsychosia
Acthalides belongs to Lemnos; this may have something to do with the supposed Tyts
rhenian origin of Pythagoras (ch. ΠῚ. n. 12).
103 elow, nn. £77078.
4 Pyrandrus is probably the same as Pyrrbus. ‘Phe skill of a certain “Delian diver”
was proverbial (DE 4.22, 902, Herondas 3.41). There iv a swimincr near the ship οἱ
Theseus as it lands in Delos, on the Peangois vaye, Perhaps sone Delian ritual lies in the
backpround.
3. Metempsychosis and “ Shamanism”
luit not Pythagoras. Obviously Heraclides first put the two traditions
tapether and the “beautiful prostitute” may be a sarcastic addition of
Hie acarchus.!5 Present throughout, however, is the assumption that
I ythaporas had “proven” his theory of metempsychosis by recounting
his previous incarnations; and also present throughout is the puzzling
“μον of “ Euphorbus.” |
a ’ ,
li Antiphanes’ comedy Neorris,! first presented not long after
442 0.C,, appears a miser who gives his slaves nothing to cat:
Ἢ 2
ivlpartos ἀνυπέρβλητος εἰς πονηρίαν,
~ ε
"τοῦτος οἷος μηδὲν εἰς τὴν οἰκίαν,
+4 T ς “-
μηδ᾽ ὧν ὁ Πυθαγόρας ἐκεῖνος ἤσθιεν
ὁ τρισμακαρίτης, εἰσφέρειν ἔξω θύμου.
He is wlluding to Pythagoras’ ascetic precepts, but the decisive word for
the imerpretation is τρισμακαρίτης. Tpis μάκαρες Δαναοὶ... (Od. 5.306)
would le a familiar phrase to every Greek, but μακαρίτης is only used
af one who has “gone to his reward,” that is, who is dead. The poet
ls puming: the “thrice blessed” Pythagoras is at the same time the
θεν dead?” One should not lay undue stress on the number 3,
whi ly has an intensive force, and is also explained by the conventional
formula τρὶς μάκαρες. But this much is certain: that it was well enough
ἀμελν in Athens, about 342 B.c., that Pythagoras had lived through
wevenil lives, that a comic poet could count on an allusion to the idea
heing understood.207
J ate testimonia often name no one but Euphorbus as an earlier
je wtiation of Pythagoras,!°8 and since this name is mentioned by both
με H Pythagoras experienced everything, he must once have been a woman. It is hard
in tell whether Clearchus was taking the matter seriously, whether he was following
Η εκ tun, or whether it was only a process of abridgment, in the course of the tradi-
finn, tliat lett hit with the same series as Dicacarchus.
. Arnipliunes fr. 168 Kock = Ath. 4.108c. The date of the Neottis can be deduced
ἤμηε π΄ τό), Ath. 6.223¢e, where allusion is made to Demosthenes’ position in the
He geet Lilies with Philip about Halonnesus in 342 b.c. (Ath. in his comment on the
Raginent, hypothesis to ps.-Demosth. 7).
ME passape is missing in the index of Kaibel’s edition of Athenaeus, s.v. Pythagoras
4" αἰ ΠΝ lor this reason) is omitted in DK and Weiher. Zeller cites it (ΤΠ 2.93), but
Mistanalites sperpeacapirns as “preiswiirdig.”” (LSJ has it right.) It is possible that
Antiptianes is dependent on Heraclides; he seems to allude to him in fr. 113 Kock
(Pdera tiles di. to ΝΜ),
oe allow. fi. tonsa PE, Died. 10.6.1, Hor. Carm, 1.28.01 (where the word iterum
gerbes ΠΟ incarnations), Nicom. (Por. VP 26 Lam. VP 63), Hippol. Ref. 1.3.3.
Av tha 195 IP De Palco, Androcydes, Eubulides, Aristoxenus, Hippobotus, and Neanthes
aie πε Capether, but it is impossible τὸ make out how much may go back to
Abltiscnin (1. 1a, with Webrli's comment p. so). That Empedocles spoke of “ Euphor-
bins (toatagai, Verbo 2goll) is unlikely; ancient scholarship on Pythagoreanisn would
hardly have tailed to mention such a {πὶ
Il PYTHAGORAS IN THE EARLIEST TRADITION
Heraclides and Dicaearchus, the conclusion has been drawn, rightly,
that this detail of the tradition is older than Heraclides.” Before the
beginning of scientific historiography, the historical consciousness of
the Greeks relates mainly to two periods, the immediate past and the
Homeric period known from the Iliad and the Odyssey; 30 that when
Pythagoras spoke of his earlier lives, he had to establish his presence in
the earlier period.4° But why did he choose precisely this not very
distinguished participant in the Trojan War? Some suggested answers
are based on the etymology of the name, but the most persuasive
interpretation is that of Karl Kerényi, who found a clue to the riddle
sn the words of Homer. As Patroclus is dying he says to Hector,"
109 Rohde, Psyche Il 418 = 599 Eng. ed.; Corssen, RhM 1912, 458. (Contra, Lévy,
Sources 34 n. 1: the naming of Euphorbus alone is a result of secondary abbreviation; but,
if that is the case, why do Aethalides, Hermotimus, and Pyrrhus never get named by
themselves as earlier incarnations of Pythagoras?)—Pythagoras recognized, in Argos, the
shield of which Menelaus had despoiled the body of Euphorbus (Diod. 10.6.2, Ov. Met,
15.163f, Schol. T ἢ. 17.28). This happens in the Heraeum near Mycenae according t@
Nicom. ap. Por. VP 27 = Iam. VP 63; cf. Paus. 2.17.3. No place is given in Hor. ge
1.28.9ff, Hippol. Ref. 1.3.3. Maximus Tyrius (10.2) speaks, erroneously, of a temple o
Athena. Heraclides (fr. 89) has Hermotimus find the shield, in the sanctuary of Apolle
at Didyma, corrupted in Tert. An. 28 to “Delphi.”” This has the appearance of being
secondary, and if so the shield story is at least older than Heraclides. Nicomachus finds
the story silly: ra yap ἱστορούμενα περὶ τῆς ἀσπίδος παρίεμεν ws πάνυ δημώδη.
110 This consideration would seem to indicate, again, that originally only one earller ἢ
incarnation, as Euphorbus, was mentioned. It may have been only later that people ᾿
calculated a certain number of years. At Th. ar. 52.88 (Androcydes, Eubulides, Aristoxenus, ©
. . 64 Ν :
Hippobotus, and Neanthes; cf. n. 108) the period is 216 years. Pythagoras himself,’
D.L. 8.14 mentions 207 years, though this is surely to be corrected to 216, which is in
itself a significant number (the cube of 6). This “ writing” of Pythagoras, cited D.L. 8.14,
ἰς to be identified with the “‘tripartitum” of D.L. 8.6f, because of its Ionic dialect (Diels,
AGP 1890, 469). See further Rohde, Psyche Il 419 = 599f Eng. ed., Corssen, RhM 1914,
243, Delatte, Vie 181, Lévy, Sources 76. O. Skutsch, CP $4 (1959) 115, conjectures ar
Ennius’ peacock originally belonged among the incarnations of Pythagoras; he wou
insert it between Euphorbus and Pythagoras. Scholars have mostly supposed that the :
source of the whole tradition was an old katabasis poem (Rohde, Q 106.1; dubiously :
Psyche II 419 = 600 Eng. ed.; Norden is very positive: it was : an ancient ee |
Pythagorean poem of the sixth century” (Vergil VI, 5 and 21) which “was regarde ao
almost canonical”; cf. Dieterich, Nekyia 129, Ganschinietz, RE X 2410. There is no reason
to believe, however, that there ever was an ancient written source, or that, if the tradition
was oral, it was transmitted in the form of a poem. What we have to reckon with is oral -
narrative and maxim (acusmata), not literary genres.
111 Delatte, Vie 1.57, interprets Euphorbus as the “good shepherd,” and assumes the
existence of a Phrygian cult of a ‘good shepherd” and an “Orphic” apocalypse deriving
from it; but this is all pure hypothesis. Skutsch, CP $4 (1959) 114, understands Euphorbus
as meaning “he who eats the right food,” and connects him with Eytiagorcah riser
regulations. It may be that both these motifs, “shepherd miysteries and correct food,
played a role; and generally it is well to keep in mind the possibility of multiple interpre
ation. .
ee ι6.8 40; Kerényi 19. Rohde mentions (Psycdie 1 418. $99 Eng. ed.) that Euphore
buy’ father was a priest of Apollo, but does not think this explains the matter adequately,
Schottlacnder (4451) cites the fact that his mother's namie was Ppovris (Η. 17.40). Corssen
(RUM τι, 22) calls the choice of Huphorbus incomprehensible ; and ‘Tert. An. all
jeered at the inationality of it
3. Metempsychosis and “ Shamanism”
᾽ ‘ a > 3 ‘ ‘ ~ Ψ cs
ἀλλά με μοῖρ᾽ ὀλοὴ καὶ “ητοῦς ἔκτανεν vids,
ἀνδρῶν δ᾽ Εὔφορβος" σὺ δέ με τρίτος ἐξεναρίζεις.
ΠῚ we consider the arithmetic here, it seems as though Moira, Apollo,
aul Euphorbus only make up two, so that two of the three must be
tilentical. The solution that Moira is not personified here, and thus not
(ΘΠ. as one of the group, is by no means self-evident to the ancient
aholars who busied themselves with the problem. “If someone
wanted to say, “Iam perhaps Apollo,’ he could, in Homeric terms,
call himself Euphorbus,”’ says Kerényi. The advantage of this interpre-
tation is that it is entirely derived from the Homeric text. The name
Kuphorbus refers unmistakably to Homer, and.the whole intellectual
world of the archaic period takes its character from Homer. Innovation
presents itself in the guise of Homeric interpretation.
We are told in fact that in Croton Pythagoras was thought to be
“the tlyperborean Apollo.” For these aspects of the Pythagoras
legend, Aristotle's book on the Pythagoreans is the important source.
We have the miraculous stories in four versions, in the Historia
μην ἐπ! of Apollonius,4® in Diogenes Laertius, Aeclian, and
lamblichus.46 Aristotle records the following items:
(a) Pythagoras was called “Hyperborean Apollo” by the Croton-
fates. !!7
(ls) At the same hour on the same day he was seen both in Croton
aiid in Mctapontum.18
"4 See the scholium on the passage.
"4 See above, ch. I 2. The material is collected by Rose as fr. 191, discussed by Corssen,
WAAL των, 20ff, and Lévy, Sources 1off. It is scarcely to be supposed that Aristotle failed
ἐλ“ πὴ the Euphorbus story, but since it was so well known he is not cited for it.
"4" The tmediate source of Apollonius is obviously Bolus, as is shown by the lemma
MuwAve at the beginning of the text (Diels, SBBln 1891, 393f; Wellmann, AbhBin 1921,
qu; levy, Sources 11.4). Theopompus (FGrHist 115F70-71) is the source of ch. § (on
reves γε) but not for ch. 6, as Diels wrongly supposes. All of it clearly comes from
Attetethe, and Bolus is, at least here, compiler rather than forger. On Apollonius see
W Kroll, RE Supp. ΙΝ 458: none of his sources is later than the second century B.c.
™ Also Plat. Numa 8, Amm. Marc. 22.16.21, and some echoes in Lucian. The wording
Hb the principal sources is often so closely similar that there may well have been a common
τ source (Bolus?). Nicomachus, though taking departure from the same
saueee, atitroduces variants that are mostly characteristic of him. He calls his authorities
audi aut ἀξιόλογοι (Por. VP 23 =- lam. VP 60).
Heil the following notes, a passage that names Aristotle is cited in bold face type.)
Ael 2.26, 0.1. 8.11, Iam. VP 140 (on the text see Deubner, SbBln 1935, 677ff; other
“πα τἀ ΕΤΟΪΚ 23, Corssen, RAM 1912, 37 τ. 1; Lévy, Sources [4 ἢ, 3); cf. lam. VP 30;
ea Pad. mort. 20.3--Nicomachus combines this with the Abaris episode, Por. VP
a8 — dain, PP 13s.
"8 Ael, 2.26, 4.17, Ap. fim. ον Its Metapontum and ‘Tauromeniuns” in Nicomachus
Vi Pay Tan. FP 194, cf 196), Phurit and Metapontum” in Philostr, VA 4.10.
pollens of Pyana imitated the feat: he was in Smyrna and Ephesus on the same day
ἡ ει, loc. eit.)
II. PYTHAGORAS IN THE EARLIEST TRADITION
(c) When Pythagoras stood up among the spectators at Olympia,
people saw that one of his thighs was of gold.
(4) He reminded Myllias of Croton that he had been King
Midas.12°
(e) He stroked a white eagle in Croton."
(f) As Pythagoras was crossing the Casas River, the river hailed him
in an audible voice, “Greetings, Pythagoras! 155
(z) As a ship was entering the harbor of Metapontum, he predicted
that a dead man would be found in 11.228.
(h) In Caulonia, he correctly predicted the appearance. of a white
bear 1533
119 Ap. H.m. 6, Ael. 2.26, 4.17, D.L. 8.11, Iam. VP 140, Plut. Numa 8. Cf. Lucian V,
auct. 6, Gall. 18, Dial. mort. 20.3. Olympia is named in Ael. 4.17, Plut. Numa 8, and Amm,
Marc. 22.16.21, and is probably to be supplied in Ael. 2.26; the other sources merely
speak of an ἀγών or θέατρον. Lévy, Sources 20, assumes that Pythagoras appeared as an
athlete (cf. above, ch. II 2, n. 5), but παρέφηνε signifies an unintentional brief disclosure,
not the nakedness of an athlete (see Plut. Praec. conj. 142c).—Nicomachus mentions the
showing of the thigh only in connection with the Abaris scene and dismisses the matter
with the verb τεθρύληται (Por VP 28 = Iam. VP 135, cf. 91f.)—~Alexander of Abonutel:
chus copied the golden thigh (Luc. Alex. 40).—On Schol. Luc. p. 124.6f, see below
n. 215.
120 Ael. 4.17, Iam. VP 143; alluded to by Nicomachus ap. Por. VP 26 = Iam. VP 63
Maddalena (359 n. 86) contests the attribution of this to Aristotle, but since the same iten
occurs in two sources in the midst of Aristotelian material, it is at least probable.—Th:
name Myllias is used in Neanthes’ horror story at FGrHist 84F31 = Iam. VP 18yff
Por. VP 61.
121 Ael. 4.17, lam. VP 142, Plut. Numa 8, Amm. Marc. 22.16.21. Nicomachus (Por
VP 25 = Iam. VP 62) shows exact verbal agreements with Plutarch (ὑπεριπτάμενω
...katayayeiv); but he makes the whole story a proof of the validity of divination by
birds, and sets the scene in Olympia, while he excludes the “golden thigh” story fron
Olympia (above, n. 119).
122 Ael. 2.26, Ap. H.m. 6. The name of the river is Kéoas in Aelian, while Apolloniu
has the meaningless κατὰ σάμον (i.e. Kdoas?), and Diogenes (8.11) Νέσσος. In the Nico:
machus tradition, Porphyry (27) has Καύκασος (meaningless; a mistake for καὶ Kdcav?
L. Bieler, WS 48 [1930] 201-205, tries to explain this last by assuming the loss of a line
in Por.; but he overlooks the context in Nicom. and the parallel in Iam., which show
that not more than one word can be wrong). Iamblichus (134) has Νέσσος (obviously
substituted, from the “handbook source,” for Καύκασος; cf. D.L., Lévy, Sources 104,
115). What we should expect is Kacas, which is a river near Metapontuin (Bacehyl,
11.119; Diels, Hermes 33 [1898] 334; Lévy, Sources 13). Can Néooos be a mishearing of
Νέαιθος, the river near Croton (as Arcerius suggested)?
128 Ap. H.m. 6, lam. VP 142, Nicom., ap. Por, VP 28 (lam, agrees exactly with Nicom,),
Cf. Thrasyllus ap. Dio Cass. 54.11.3, CCAG VIIE 4.100. 1fF.
124 Ap. H.m. 6, with a gap which is filled from fam. VP 142. Nicomachus has a different
version (Por. VP.23° lam. VP 60), according to which, in Daunia, Pythagoras converted
a wild she-bear τὸ vegetarianism, and a similar tale (Por, PP2g 0 Jam. TP Or) about an
ox in Tarentum which, on the urging of Pythagoras abstained for the rest of tts lite frog
cating: beats.
3. Metempsychosis and “ Shamanism”
(1) In Etruria, he bit a poisonous snake to death.!*
(1) After predicting to the Pythagoreans the outbreak of civil strife
(εν...) he disappeared to Metapontum without anyone’s seeing him
yo l8
Iwo more reports are only found in Jamblichus, in context with
the Aristotelian material. It is possible that they go back to Aristotle:
(k) Pythagoras took from Abaris, the priest of Apollo from the
cuuntry of the Hyperboreans, the arrow with which he traveled, and
thus established himself as the Hyperborean Apollo?”
(I) “They say of the man who bought Pythagoras’ house and tore it
awn, that he did not dare tell anyone what he saw, but that as a
remult of this crime he was convicted of sacrilege by the Crotonians
andl executed. For he was convicted of having stolen the golden chin
which had fallen from the god’s statue.’ Pythagoras’ house is inviolate,
like a sanctuary of the mysteries; the transgressor dies the death of a
aeapepoat.!8
‘These “miracles”’ are portents without interpretations, revelation
aml oecultation at once. At a certain moment there is a glimpse of the
divine the gleam of the golden thigh, the greeting of the river god,
the arrival of the Hyperborean. Superhuman powers are evident in
Vythaporas’ prophecies, in his mastery of the animals, and in his
camtrol of space and time, as well as in the numinous dread that
4 Ap. Han. 6. The tone is different in Iamblichus (142): τὸν μικρὸν ὄφιν, ὃς ἀπέκτεινε
Admit Iase corrects ὅς to ὄν, but probably Iamblichus himself changed the wording.
Ihe passes on, with the phrase ὁμοίως δέ, to the story of the driving out (ἀπεπέμψατο) of
ἀν ἢ stake from Sybaris. (It is doubtful whether the double version stems from
Asiatotle.) Uhae Pythagoras himself bit the snake (like must be combated by like!) was
bart protesque for lamblichus.
Ag, Than. 6.
Lam. EP tgof, excluded from the Aristotle fragment by Rose. Corssen, RM 1912,
iH, αν to show that Hermippus is its source, using the argument that Pythagoras, who
ainply takes his arrow away from “the unfortunate Abaris,” is presented in an
uativonable lighe; but, if Pythagoras is the “‘Hyperborean Apollo,” Abaris has reached his
He 1 anc does not need the arrow any more (Lévy, Sources 18). Lévy, (Sources 14f)
πεν νι Chat the rare feminine ἡ οἰστός occurs not only in section 140 but also, pre-
rtody, a Aristotle (Phys. 239b7). The excursus on Abaris, sec. 141, p. 79.18-23, has a
thee αὐ ἀπ Πρ to Ap. Π μην 4, and probably comes from a different source from the
ΘΠ Παρ, context. In that case, lamblichus may well be the author of the transitional
pore pp. 7... 80.0. The expression ὁμολογοῦντα ἐποίησεν, the “confession” of Abaris,
aa ΠΝ ΠΕ of the confession of Peter after the transfiguration of Christ, can scarcely
με ἢ} Perhaps it isnot coincidence that in this passage (p. 79.236) οὐστός is twice masculine
“πα deminine, in the manuscripts. (Deubner restores the feminine throughout.) The
parallel version of the Abaris story ace hin. PP ortPhas only the masculine.
han EP ogg. Pimacus reports that Pythagoras’ house became a shrine of Demeter
(ἐν ἢν 5 40}}.11}.. On scapegoat rituals see Ve Gebhard, REV A 12goff.
ll. PYTHAGORAS IN THE EARLIEST TRADITION
attaches to his house. There is always something enigmatic about the
meaning of these miracles, which is apparently revealed to the insider
but not explained to the uninitiated. The relation to Apollo is ambigu-
ous, too. According to Aristotle, the Pythagoreans taught, ἐν τοῖς πάνυ
ἀπορρήτοις, that “among rational beings there is that which is god,
that which is man, and that which is like Pythagoras.””!?° Here a phrase
like λογικὸν ζῷον betrays later terminology, though the idea of the
intermediate would not contradict the identification with “Hyper
borean Apollo” : the third kind of being could be gods in human form,
But explicitness is avoided, just as the Euphorbus story does not go
beyond a ‘“‘perhaps”’ as to the hero’s divine status.
A second group of legends is also attested in fourth-century sources,
but it was controversial even then whether they apply to Pythagoras
or Pherecydes of Syros. In his Tripus, a book about the Seven Sages,
Andron of Ephesus ascribed to Pythagoras what Theopompus ascribed
to Pherecydes. Porphyry, who brings out the contradiction, brands
Theopompus the “thief ”8°—which prompts modern scholars to take
the opposite view.!5! These are the stories:
(a) Either Pythagoras in Metapontum or Pherecydes in Syros!™ took
a drink of water from a well and predicted an impending earthquake.
(b) Pythagoras in Megara Hyblaea or Pherecydes in Samos predicted
that a ship, sailing with a favorable wind, would sink, and this
immediately happened.
(c) Pythagoras predicted the conquest of Sybaris or Pherecydes that
of Messene, and in each case a friend was warned.
(4) Pherecydes or Pythagoras, on the authority of a dream in which
Heracles appeared, told the Lacedaemonians not to honor gold of
silver; and in the same night Heracles bade the kings to heed the words
of Pherecydes or Pythagoras.’
129 Arist. fr. 192 = Iam. VP 31; cf. Iam. VP 30, 143f, Schol. BT II. 1.340. Ps.-Apollas«
nius of Tyana, Ep. 50, counts Pythagoras ἐν γένει δαιμόνων. Corssen (RAM 1912, 39) and
Kerényi (18) emphasize that this is not incompatible with the equation Pythagoras-Apallo,
180 Porphyry, Φιλόλογος ἀκρόασις ap. Euseb. Praep. evang. 10.3.6 (DK 7A6)
Theopompus’ version is also attested at ID.L. 1.116 and Ap. H.m. 5 (FGrHist 11$F70
71), Andron’s (earthquake and sinking ship) at lam. VP 136.
131 For priority of the Pherecydes version are Corssen, RIM 1912, 3 3ff, Bertermani
$8f, Rathmann 28.
132 ἴῃ Samos, according to Max. Tyr. 13.5. No location is named in Cic, Div. 1.61
and Plin. HN 2.191.
138 This story is only found in D.L. 1.112. Theopompus P71, with the addition ὄνιθ
δὲ Πυθαγόρᾳ περιάπτουσι ταῦτα. But Porphyry mentions that Andron reported atil
tore miracles of Pythagoras (καὶ ἕτερά τινὰ τούτοις ἐπαγαγών). Andron namet
Pherecydes as Pythagoras’ teacher (ILL. 1.110)
3. Metampsychosis and “ Shamanism”
‘These miracles all belong to the realm of prophecy. Prediction
wl an carthquake is also ascribed to Anaximander; the addition
wi the “drink of water’’ suggests an origin in cult practices.85 Com-
jutison of details! yields no conclusive argument for the priority of
either version, though there is a serious chronological difficulty in
the juxtaposition of Pherecydes and the fall of Messenia.1*? It is scarcely
pusstble to judge the date or the reliability of Andron; but there is no
tkoult that Theopompus was hostile to the Academy and also to
Vythaporas.138 In any case Andron attests the wide dissemination of
Hythaporas legends in the fourth century. There is no reason to think
al Aristotle as being dependent on Andron.189
later tradition adds little beyond the miraculous catch of fish,
which docs seem to be earlier than Nicomachus.4° What Aristotle
feports is in tone, quite apart from the nature of its attestation, anything
but characteristic of “late antiquity.” It was more likely to provoke
later apes to mockery or protest. Alexander of Abonuteichus claimed
6}. Anaximander, see Cic. Div. 1.112, Plin. HN 2.191 (DK 12.4.54). The mention
af Ataxaporas (Amm. Marc. 22.16.22 [DK 59Aro]) is doubtless the result of confusing
the two men.
@ At the oracle at Clarus, the priest prophesied after drawing water from a secret
oping ina cave (Tac. Ann. 2.54; cf. M. Ninck, Die Bedeutung des Wassers im Kult und
fehen der Alten [Leipzig, 1921] 83ff)—Apollonian prophecy is combined here with
"pliysical”’ speculation about the subterranean forces that cause earthquakes.
ΜΝ (ἢ (RAM 1912, 36) stresses that Pythagoras is not ordinarily connected with
Metly (Megara Hyblaea), But why would a plagiarist not have put down “Croton” or
Metspontum,” rather than betray himself? Aristotle has Pythagoras traveling to
Htruria Gubove, ἢ. 125).
ΜΕ ΟἽ Kiechle 11 and 56. The fall of Messene is set at 600 B.c. (at the latest) in ancient
μην, but Pherecydes was a contemporary of Anaximander and perhaps dependent
ΝῊ lita (von Fritz, RE XIX 2030f), though some authorities give an earlier date (Arist.
ἢ 21, Suda s.v.).
442. 4; sce above, ch. II 2, n. 61. This counterbalances the principal argument for the
preity of the Pherecydes version, namely that the attribution is more likely to have been
gwiteliedl in the direction of the more prominent figure, Pythagoras.
1 Ay 1 οἰνάς (Q 135) thought. Corssen (RhM 1912, 32ff) equates the second anecdote
given by Andron and the similar one in Aristotle (above, n. 123), and denies Aristotelian
aithoralup. Porphyry’s wording makes it clear, however, that Andron told the same story
4¢ Eheopompus (ID.L. 1.116), namely that the ship sank, whereas in Aristotle it has a
ἢ board. Corssen finds Theopompus’ version simpler, but one can also interpret
ἢ, with ws sensational tone, as a cruder variant of Aristotle’s.
4 Mico, ap. Por. VP 25 == lam. VP 36; somewhat differently Plut. Quaest. conv.
ΒΜ. “9.44, De cap. ex inim. ut. 9.g1c, Apul. Apol. 31. According to Nicomachus the event
δι. place between Sybaris and Croton, according to Apuleius near Metapontum.
Pythapores cither buys some fishermen’s catch and sets the fish free (Plut., Apul.), or
predicts the number of fish that will be in the net and then, when the prediction turns out
«4, bids the fishermen set the fish free. Knowing a number in advance is a feat
ΗΠ {τὸ the seer from carliest times (Hes. Melampodia fr. 278 M.-W.); but here the
edit of prophecy could have been an addition of Nicomachus himself.
IL PYTHAGORAS IN THE EARLIEST TRADI TION
to be Pythagoras and displayed his golden thigh. In Lucian’s judgment
he selected and even excecded “all the vilest and most damaging
slanders ever vented against Pythagoras.’’4" The stories of miracles are
widely regarded as attempts to discredit him and put down 7 slander.
Nicomachus expresses candidly his distaste for the crude, plebeian
miracles42 and tells the stories in such a way that the inexplicable is
played down or ignored, so that what remains is, essentially, clair-
voyance and the doctrine of the kinship of all living beings, and the
exhortation to ἡμερότης. The golden thigh is only mentioned in con-
nection with the wonder-worker Abaris, and is quickly disposed of,
and completely unsophisticated details like appearing simultaneously
at two places or biting a snake to death are omitted. In their place
appear the somewhat sentimental tales about the conversion of wild
animals. But even much earlier than Nicomachus these reports had been
transformed. “‘Epimenides, Eudoxus, and Xenocrates” are cited as
saying that Pythagoras was the son of Apollo.“ This eliminates not
only metempsychosis but identity with Apollo, and closeness to the
gods is expressed in the customary form of divine ancestry.
The Pythagoras legend is attested for the fourth century B.C., and
at least part of it was well known; what is more, it 1s distinct from the
Platonizing interpretation current in the Academy, which made
Pythagoras a doublet of Plato himself. It antedates Antiphanes,
Andron, Heraclides. It was recorded by Aristotle, perhaps not without
a polemical glance toward the Academy’s modernizing interpretation,
It was carried along, willy nilly, by others, though assiduously rcine
terpreted or, sometimes, rejected as slanderous, and all the while made
the target of ridicule by its enemies. Finally, then, with figures like
Apollonius of Tyana and Alexander of Abonuteichus, the wonders
ἊΨ Ὁ « Gee ae +t
worker reappears in reality; such activity 1s to be sure “‘late antique,
but it revives preclassical patterns.’
141 Lucian, Alex. 4 (cf. 40).
142 n. 109, 119, 117, 121, 124. "
349 oe = sae et fr. 22 H., Eudoxus fr. 86 Gisinger τ: fr. 324 Pee ὧν
authenticity of the Xenocrates fragment is doubted by Rohde (Q 128), Zeller ᾿ iene ok
and Lévy (Sources 9.5), that of the Eudoxus fragment by Schaarschmict oe Hee He
adequate grounds. Xenocrates had to reinterpret: the Pythagoras ee (a ue
19-20). Apollonius (lam. VP § Por. VP 2) quotes two lines from ane epy ΣΝ ᾿ Ἦν
poct” in which Apollo is named as Pythagoras father. Lévy (Sources, es . 4, Ἤν
n. 4) thinks the citation of Epimenides, Xenocrates, and Fudoxus i an pas a eas
Apollonius passage; if correct, iat baddies speak for its authenticity, for in that case
tohave been taken froma “handbook = source.
ma i influence of the Pythagoras legend is traced by Lévy, La. cf. above, 1, "" an
its effect on Athanasius lite of St. Authony, nee ΙΧ, Ἰλοἰἰσύμμοιι, age ers Ι, Ἢ
Lég. ὙΠ Lévy, however, stronply overrated the influence ol Heraclides on the legend.
3. Metempsychosis and “ Shamanism”
Oral tradition clearly is involved in the Pythagoras legend; we must
expect alterations and distortions to have occurred. Still, for its origin
li terminus post quem is set other than the historical Pythagoras,’
aud faithful preservation of original traits cannot be excluded from
the start. This is why we should try to understand the tradition before
ihiscarding it, even though, consciously or not, scholarship tends to
proceed from the assumption that legend is always secondary, because
the “historical kernel’’ must be an event or pattern of events amenable
to common sense and not a “‘miracle.”” What if the “facts,”
ws we sce them, were experienced differently at that time? If
the historical Pythagoras taught metempsychosis, this same _hist-
wtical Pythagoras must have claimed superhuman wisdom, he had
(0 tse his own life as an example and find himself in the Trojan
War. And if he wanted to make this credible, he had to—perform
Miracles.
Vhe katabasis of Pythagoras is very iniportant in this context, but
it is especially difficult to evaluate this tradition. Therefore we may first
ΜΠ νον what there is of similar phenomena in archaic Greece, in the
hope that this will bring various kinds of confirmation and illumina-
then.
Irom ancient times Pythagoras, as miracle-worker, has been asso-
εἰ] with figures like Aristeas, Abaris, Epimenides, Phormio, and
Hn podocles.146
An cpic poem entitled Arimaspeia, by Aristeas of Proconnesus,
8 Ty all cases of coincidence with the stories of Aristeas, Epimenides, or others, Rath-
Hann assumes that the ascription to Pythagoras is secondary; and Corssen (RAM 1912,
44) beheves that the motif of simultaneous presence in two places (above, ἢ. 130) was
ayyueprtuted for Pythagoras from a version of the Aristeas story not known to Herodotus
‘heliw, on. 148). But if everything is late ascription to Pythagoras, where did his fame
tenmie drana? How could a cipher have exercised so much attraction?
‘Ap. lm. 1-6: Epimenides, Aristeas, Hermotimus, Abaris, Pherecydes, Pythagoras;
Pleas Al Strom. 1.133.2 (cf. Tatian 41): Pythagoras, Abaris, Aristeas, Epimenides,
fianaater, Enapedocles, Phormio; Nicom. (Por. VP 29 = Iam. VP 135): Empedocles,
Vyumetudes, Abaris, Pythagoras; Max. Tyr. 10.1ff: Epimenides, Pythagoras, Aristeas;
Vlas #2N 7.174: Hermotimus, Aristeas, Epimenides, Empedocles; Procl. In Remp. Il 113:
Atiateas, Elermodorus (cf. below, n. 177), Epimenides; Greg. Naz. (Migne 35.581, cf.
ye tayt, $8.47): Empedocles, Aristacus (i.c. Aristeas), Empedotimus, Trophonius. As
catly as Democritus’ book Περὶ τῶν ἐν “Aidou there was a treatment περὶ τῶν ἀποθανεῖν
Ἰμξήίνεων ἔπειτα ἀναβιούντων (DK 68131). Wellmann (AbnBin 1921.4, 12f) thought this
nok was a forgery by Bolus; bue it was included in Thrasyllus’ tetralogically arranged
dtulopue of Democritus’ writings; certainly spurious items, like Bolus’ Xeipéxpyra,
pps wan appendix to the catalogue (D.L. 9.49). ‘The topic was current in the fifth
Hy δ. Οὐ as is shown by Soph, EL 62}
Π. PYTHAGORAS IN THE EARLIEST TRADITION
was in circulation in the early sixth century B.c.14” Aristeas told how,
possessed by Apollo (φοιβόλαμπτος γενόμενος) he had traveled to the
country of the Issedones in the far north, and learned from them about
the Arimaspi, the griffins, and the Hyperboreans who lived still
further north. Herodotus adds a local legend from Proconnesus, to the
effect that Aristeas died, and soon after was seen traveling abroad,
while his body was found to have vanished. After seven years he
appeared in town again, bringing his Arimaspeia, and then disappeared
again. Herodotus also was told, in Metapontum, that Aristeas had
appeared there and bidden the natives to build an altar to Apollo,
“For (he said) Apollo had visited them alone among the Italians, and he
himself had accompanied him in the form ofa raven ... And even now
there stands in the agora, near the statue of Apollo, a statue inscribed
with the name of Aristeas, and there are laurel bushes round about.’"™4#
Herodotus calculates that the appearance of Aristeas in Metapontum
occurred 240 years after his disappearance in Proconnesus. It is hard
to account for this dating, though the incident in Metapontum must
have been relatively late; one is tempted to think that the Metapontine
coins might be relevant which, beginning to appear about 470 B.C,
show Apollo with a branch of laurel.’°°
The report from Proconnesus is inconsistent. First Aristeas dies, ther
his corpse disappears; possession and disappearance stand side by side
147 J, D. P. Bolton, Aristeas of Proconnesus (Oxford, 1962; cf. Gnomon 3.5 [1963] 23 5-240)
The terminus ante quem is a representation of Arimaspi on a mirror from Kelerime:
(Bolton, pl. 1, ca. 575 B.C.), the terminus post quem the mention of the Cimmerlat
invasion. Pindar mentions Aristeas (fr. 271), and the Arimaspea was used by Hecatacu
(Jacoby’s notes on FGrHist 1F193—4) and by Aeschylus in the Prometheus (Bolton 45-64)
The most important evidence: Hdt. 4.13-15.
148 Between Artace and Cyzicus, according to Hdt. 4.14, “‘on the way to Croton” lt
Plut. Rom. 28, a contamination with the story which follows in Hdt.
149 Hdt. 4.15. At Ath. 13.605c, too, there is a mention of the δάφνη χαλκῆ (δάφναι lt
Hat.; cf. Giannelli 63 ἡ. 1) of Aristeas in the marketplace of Metapontum. (Perhaps frou
Theopompus: see Jacoby, FGrHist 115F248 n.)
150 Giannelli 62 (cf. Head? 76; 5. P. Noe, The Coinage of Metapontum Il [New York
1931] nos. 314ff, on the dating II gff. One coin shows an altar next to Apollo, no. 319.)
Herodotus at least places Aristeas later than Homer (2.53; according to Strabo 14.639
“some” would reverse the relationship). Proconnesus was founded in the time of Gyge
(Strabo 13.587, $90). If the connection with the coins is correct, and Herodotus thu
learned in Mctapontum, about 440, about an event of approximately a gencration carlict
the Arimaspea would be dated about 710; but this docs not fit the history of Proconnesy
(sce also Bolton [above, n. 147] 127). All the same, we must reject E. Schwyzer's con
jecture of συγκυρήσας for συγκυρήσαντα, which would show Herodotus dating his owi
visit to Metapontum rather than the appearance of Aristeas (Ph W 42 [1.22] 428; tollowe
by Schinid [1.303 9. 3). The point Herodotus is making is the time between Aristeat
two apparitions (Mculi 1 $4.2).
3. Metempsychosis and “ Shamanism”
In st that “Herodotus has combined two versions of the legend’’?!5!
| ater reports are unequivocal: the soul leaves the body and hovers about
in the air “in the form of a διγά. 155 Perhaps the contradiction itself,
the failure to smooth over difficulties, and the lack of a clear separation
uf body and soul, are signs of an archaic way of thinking.
Aristeas is connected with Pythagoras by more than the similarity
if the legends in which they figured.153 Pythagoras was regarded by the
initiated as the Hyperborean Apollo and died in Metapontum near the
lepinning of the fifth century. But the main authority on the Hyper-
luireans was Aristeas, and the god who flew about the world, accom-
puinicd by his servant in the form of a raven, and finally alit in his
apectal city, was the Hyperborean Apollo. If, about 470, a mysterious
prophet brought it about, by his message, that an altar and a statue were
set tip, then the new cult was obviously dedicated to the Hyperborean
Apollo, and for whatever Pythagoreans there were in Metapontum
at that time we must assume that the proclamation of Aristeas was
niderstood, at least ἐν ἀπορρήτοις, as meaning that Apollo had lived
jn Metapontum in the form of Pythagoras.5 The cult belongs
with the Icgend; in both is reflected the activity of Pythagorean
gircles.
Abaris, too, is a priest of the Hyperborean Apollo, and even comes,
linwelf, from the land of the Hyperboreans. He brought gifts from
4) Jolie H 92.1 = 329 ἢ. 109 Eng. ed.; Corssen, RhM 1912, 44.
4% Max. Tyr. 10.2, Plin. HN 7.174, Suda s.v. Aristeas.
"8 According to a late version (Ap. H.m. 2) Aristeas was seen on the same day in
Hrouwennesus and Sicily. (For Pythagoras, above, ἢ. 118.)
"δ. The carliest mention of the Hyperboreans: Epigoni fr. 3, Hes. fr. 150.21 M.-W.;
the αν] of Apollo in Delphi from the land of the Hyperboreans: Alcaeus fr. 307
il, wanda shield strap from Olympia (600-575 B.c.) in E. Kunze, Olympische Forschungen
" (iuso) 74; further references to the flying Apollo: Gnomon 35 (1963) 239 n. I. The
titeehs associated the name Hyperborean with Βορέας and Bopeddes, and this fits in well
with the disappearance stories (Hdt. 4.36, Diod. 2.47.7 = Hecataeus of Abdera FGrHist
4641) J. Hlarmatta has recently suggested a derivation from *8épis, ‘mountain”’
(Arta ἐπ. 3 [1955] 5684); A. J. van Windekens favors derivation from the root of φέρειν
Csemductor”’ into the next world?), RhM too (1957) 164-169.—According to Ath. 605c
(alueve, un. 149), Aristeas comes to Metapontuin “from the Hyperboreans.”’
"ef Bolton 174f. The connection between Aristeas and the Pythagoreans was
εὐ νοι by Rohde, Psyche I 99.2 -= 333 n. 122 Eng. ed.; Diels, Parm. 21; Giannelli
fy G4 “Aristeas of Metapontum” is found in the catalogue of Pythagoreans, lam. VP
p t442 Deubner. According to Iam. VP 138, the Pythagoreans believe everything that
ja sath about Aristeas and Abaris.—Stein, in his note on Hdt. 4.15, interprets the visit of
Apollo with relation to the remarkable fertility of the land of Metapontum; but from
the very beginning this was represented by the car of grain on the city’s coins, Εἰ Pais,
Siudndella Sicilia L804) $48, conjectured, arbitrarily, that Herodotus had misunderstood,
aul that Aristacus rather than Aristeas had been Apollo's paredros in’ Metapontum,
Il. PYTHAGORAS IN THE EARLIEST TRADITION
there, according to Attic legend, to the Proerosia festival at Eleusis.1&
Abaris has connections with other Greek cults,5’ and there were
charms and oracles attributed to him.15* Probably his meeting with
Pythagoras had been recounted even before Aristotle’s time,!®* though
there were chronological difficulties.®°
According to Herodotus, without ever eating, Abaris carried
Apollo’s arrow all over the world,!* but as early as Heraclides it was
said that he flew on this arrow,!® and this version is regarded as the
original one.1* It may be, though, that just as in the case of Aristeas,
the tradition was selfcontradictory from the beginning. There were
alternative ways to report the activities of the miracle-worker; Abaris
could not perhaps “actually” fly, but he could claim the ability, and
even, in ecstatic ritual, act it out, as 1t were, as a shaman. Whoever was
ready in his heart to believe, would speak of “‘flying;”’ those who were
ready to discard the old-time magic would report the matter in the
style of Herodotus.
Epimenides, the famous Cretan “purifier,” comes from a different
environment. That he slept for decades in the cave of Zeus was known
already to Xenophanes.1* This puts Epimenides in the main line of
specifically Cretan cult and myth. King Minos visited his father Zeus
in that cave every eight years.'® Epimenides called himself Aeacus—
156 Lycurgus frr. 84-85 Blass, Hippostratus FGrHist 568F4; cf. the Hyperboreans in
Delos, Hdt. 4.33.
167 Sparta, Kore Soteira (Paus. 3.13.2, Ap. H.m. 4, Iam. VP 92, 141); Cnossus, (Iam,
VP 92); Palladium (Firm. Mat. Err. prof. rel. 15).
158 ἐπῳδαί, Pl, Charm. 158b, where Zalmoxis is also mentioned; Ap. H.m. 4, Schol. Ar,
Eq. 729.
159 Cf. n. 127. On Iam. VP 215, ch. I 1, n. 13. The source of the exposition in Iam. VP
91-93 cannot be determined (Heraclides?). Nicomachus (lam. VP 135 = Por. VP 2)
presupposes it.—Abaris is given as a Pythagorean in Iamblichus’ catalogue, VP p. 145.17,
160 Pindar (fr. 270) puts him into the age of Croesus, Hippostratus (FGrHist 568F4) at
568/565, “others” at 696/693. Cf. Jacoby on 568F4.
161 Hdt. 4.36. Lycurgus, too, says that Abaris carried the arrow (fr. 85 Blass == Harpos
cration s.v. Abaris), as does Aristotle (lam. VP 140; above, n. 127). For both lingtaisele
and material reasons, we must reject the conjecture, repeatedly proposed (c.g. M. Maye
RE XV 1357f, Roscher, Lex. 1 2837) to substitute ὡς τὸν (=. τοῦτον) ὀϊστὸς περιέφε
for ὡς τὸν ὀϊοτὸν περιέφερε in the text of Herodotus.
162 Heraclides fr. «τς. He is αἰθροβάτης in Nicomachus (Por. VP 20 - - Iam. VP 13
and in Iam. VP οἱ.
163 Corssen, RUM 1912, 47, Meuli 1syf, Dodds, Irr, 161 πὶ 33 (contra Rohde, Psyche |
gn. 1 == 327 n. 108 Eng. ed.).
164 Xenophanes DK 21 B20 (Epimenides lived to the age of 164). Theopompus explicit!
mentions the Dictacan cave (PGrHlist 15267-6090, Max. ‘Tyr. to.t; cb 37-0), though ene
might be more disposed to think of the one on Tda, mentioned by ‘ Epimenides” (DK
3824; Rohde, Psyche Dizon. α rok a. 24 Eng. οὐ).
16> Od, 19.078, [PL] Minos qioe, PL Leg. 6240, On the Cretan Zeus, Nilsson | 42aff.
3. Metempsychosis and “ Shamanism”
this making himself a brother to Minos—and claimed “that he had
heen reborn many times” (πολλάκις ἀναβεβιωκέναι).166 The Cretans
valle him νέος Κούρης, thus bringing him into close relationship
with Zcus.'!87 A voice speaks to him from heaven;1* and the Cretans
aa titice Co him as to a god.169
Oracles and theogonic poems were in circulation, bearing the name
wl Epimenides.7® Doubtless much was attributed to him by later
fingers, Plato’s report that Epimenides prophesied the Persian War
ten years before it occurred is explained as meaning that at that time
a new book of oracles ‘by Epimenides” appeared?! Legendary
jatits were added, too. Nevertheless, the purification of Athens from
the (cylonian curse in the time of Solon may be regarded as historical.172
There were “caves of Zeus” in Crete, sites of the initiatory ceremonies
al wecret societics, who are reflected in myth as “Dactyls”’ or “ Kou-
fetes," "The initiation of the καθαρτής by sleeping in the cave of Zeus
le comprehensible from this point of view. It also represents death and
feairrection, for in the meantime Epimenides was regarded as dead.
‘The long fast, made possible by the magic food ἄλιμον, as well as the
faltiwinp, also are bound up with ritual. But given these facts, his
being “reborn many times” and his identification with Acacus are not
Necessarily “purely Pythagorean fabrication.’!”* There is never any
Wention of a metempsychosis doctrine of Epimenides, only the unique
eharacter of the initiate. The explanation is rather to be found in a
Medd satg: cf Procl. In Remp. Il 113. When the Suda, s.v. Epimenides, says ὡς ἐξίοι
ἡ ψυκὴ dinar ἤθελε καιρὸν καὶ πάλιν εἰσήει ἐν τῷ σώματι, the precise wording is no doubt
φρενί! γε perhaps taken over from Aristeas (as Dodds, Irr. 163.42); but it is hardly original
fe tle see either.
? Myronianus ap. D.L. 1.115, Plut. Solon 12. Cf. the hymn to Zeus from Palaikastro,
in which the pod is addressed as έγιστε Kodpe Kpdvece (Nilsson I 3226).
© Uheopompus (hGrHist 115F69, D.L. 1.115): as Epimenides was outfitting a sanctuary
ty the Nytuphs, a voice commanded, μὴ Νυμφῶν ἀλλὰ Διός. This Zeus was, then,
wrpaliiped ina proteo or cave.
69. 1} 2.114.
“ἢ llected in DK γ1}, and FGrHist 457. See esp. Diels, ΘΒΒίη 1891, 387ff.
PENT Pee ἡ δες Diels, SBBIn 1891, 395, and DK I 32 n.
"a ΑΙ Uth, Pol 1, Plut. Sefon 12, Neanthes FGrHist 84F16 (Ath. 13.602c). See Diels
RAMs «με, 1ὼ0.Ψ
8 An: Labo τῷ, Por. [Π)}Ὀ 17, Bur. fr. 472. Cf Nilsson I 261-264; P. Faure, Fonctions des
rabeties civtatves (1904).
MEM por Theophr. Hist. pho 7.1 2.1. D.L. 1.114, Plut. Conv. sept. sap. 1§7d (DK 34).
Vleet the as attributed to Heracles by Herodoras (FGrHist 31F1), to Pythagoras by
PNogenes Antonius (Por, bP 34). See Rohde, Rem. 275 n., Haussleiter Τοῦ Dodds, Irr.
ia ge (els regarded the δ Orphic asceticism” of Epimenides as secondary, SBBIn
Hut, 191) On tattooing, sce Sudas.v. Epimenides (DK 3A2), Dodds, irr, 163 nn. 43-44,
He 0] ν SUN akot, WON, 1.
U. PYTHAGORAS IN THE EARLIEST TRADITION
parallel development from common origins; Pythagoras too, as the
legend says, sought initiation in the cave on [44.175
A journey of the soul, in pure form, was attributed to Hermotimus
of Clazomenae: His soul left his body and wandered about, while the
body lay as though dead, until one day his enemies burned the body
while the soul was absent’? Hermotimus could predict future events;
the Clazomenians built a sanctuary in his honor.!”* Here too there is
ritual to go with the legend.
The stories of Phormio and Leonymus take us to Croton. Phormio is
mentioned as early as Cratinus, interestingly enough in the comedy
Trophonius. Theopompus relates, among other things, that Phormie
was wounded, in a battle, by divine opponents, the Dioscuri. He
was told by an oracle to travel to Sparta—obviously to the house
of the Dioscuri; and when he laid hand on the door, he found
st was the door of his own house in Croton. He was home, safe and
sound.}79 ᾿
The tale of Leonymus is almost like a doublet of this.18° In the
battle on the Sagras he was wounded by Ajax, who was fighting in the
ranks of the Locrians. To be healed he had to go, at the behest of the
Delphic Oracle, to the “ White Isle,” where he met Achilles and Ajax,
116 Por, VP 17 (Antonius Diogenes?), D.L. 8.3. The connection of Pythagoras and
Epimenides is obviously secondary; Pythagoras is sometimes the teacher (Nicom., Por,
VP 29 = Iam. VP 135; cf. lam. VP 104, 221f) and sometimes the pupil (Apul. Flor, τ4 py
59, D.L. 8.3). For a purported piece of evidence from Epimenides about Pythagoras, 9498
above, ἢ. 143.—A unique rebirth story was told of Aesop, as early as the fifth century Βιθι
(Plat. Com. fr. 68, ca. 400 B.c.; Schmid I 4. 145 Hermippus ap. Plut. Solon 6; Ptolemaeus
son of Hephaestion ap. Phot. Bibl. 1§2b11). This is not “parody of Pythagorean
teaching” {as Hausrath says, RE VI 1710; cf. Schmid I 1.675 n. 2), but has its roots
in ritual. The uncomfortable feeling about the killing of the scapegoat was countere
balanced by the purported revival (A. Wiechers, Asop in Delphi [Meisenheim, 1y61}
ff).
pe Ap. H.m. 3 (perhaps from Theopompus, like sections 1 and 5; Rohde, Psyche il
95.1 = 331 ἢ. 112 Eng. ed.), Plin. HN 7.174, Plut. De gen. 592c-e (“Hermodorus"),
Tert. An. 44.
178 Ap. H.m. 3, Tert. An. 44. According to Aristotle, Hermotimus had a doctrine about
νοῦς before Anaxagoras (Met. 984b15; fr. 61 = Iam. Protr. 48.16f) cf. M. Detienne, “ΠῚ
origines religieuses de la notion de l’intellect: Hermotime ct Anaxagore,” Rev. Philos,
89 (1964) 167-178.
179 See Meineke, FCG II 1227ff; Diels, Parm. 17ff; Cratinus ft. 223 Kock; Theopompuf
FGrHist 115F392 -- Suda s.v. Phormio (who adds an ecstatic journey to Cyrene). Clean.
Al. Strom. 1.133.2 associates Phormio with Aristeas and the rest (above, πὶ 146); Paus. 1
gives a different story of Phormio the Spartan and the house of th. Dioscuri.
180 Pous, 3.19.11-13. Somewhat differently Conon PGrH fist 26F 1818, who has no epiply
any on the island, only a sacrifice. Το, An. 46.9 and Herm. Phd. p. 74 Couve, κα
Schol. PL Phdr. 241. have him healed in his sleep. On the matter of the battle at eh
Sagras, above, ch. ΠῚ 2, 1. 40.
3. Metempsychosis and “ Shamanism”?
wil then returned home sound. The “ White Isle,’2® later localized
nthe Black Sea, was originally identical with the “White Rock” in
lhe underworld, and Leonymus was “the first” to visit this island.
I huis in this case recovery is not possible without a regular journey to
lhe nether world. Phormio obviously did his “traveling” in a trance
late; and many thought that Leonymus had only been dreaming.
‘The legend of Stesichorus is bound up with the story of Leonymus:
the latter brought from the White Isle Helen’s instructions to
Stesichorus, that by composing his palinode he might regain his sight.
Noth the Leonymus and the Stesichorus stories have been supposed
ti po back to a Pythagorean origin,!® and a certain amount of coinci-
lence in place and time is not to be denied, though the epoch of
Ntesichorus is earlier than that of Pythagoras.18* There was a belief
iv southern Italy that healing could be won by an ecstatic journey into
the world beyond, to the gods. And as they thought of their ancestral
giuls as quite literally fighting in the ranks,!** the healing legends too
wight be bound up with ritual activity.
ln {ifth-century Sicily, Empedocles could promise his pupils: “Thou
ahalt stay the power of the unwearied winds which sweep upon the
earth... and, if thou wish, bring back their breath again. After dark
twin thou shalt cause a seasonable drought for men, and after summer's
iroupht bring on the streams that nourish the trees... Thou shalt
ἐγ, back from Hades the strength of a man who has died.’’18° The
Hinpedocles legend reports the miracles as having been accomplished;
as lias been said, he was “the creator of his own legend.’’18* But how
sould he have created it, how could he have called himself a god, if
ls was not able actually to perform, or at least to pretend to perform
1 The translation of Achilles to the “ White Isle”’ is already mentioned in the Aethiopis
(Wiewl Cchrestom.). The localization of the island in the Black Sea is as early as Pi. Nem.
440 Cl Rohde, Psyche IT 371ff = 5§37ff Eng. ed., Roscher, Lex. s.v. Leuke.—Od. 24.11.
Weticnne, RHR 1957, 129ff. Diels, Parm. 17ff, suggested that an epic poem on the
hatele wt the Sagras was the source of the story of Phormio and Leonymus.
4 The new fragments of Stesichorus (D. L. Page, Lyrica graeca selecta [1968] 263-268),
along with the archacological evidence, confirm the dating of Stesichorus in the first half
ἡ απ sixth century.
64 (κα the battle against Sybaris, Milo led the Crotoniates in the garb of Heracles (Diod.
ει 9.0). Croton was regarded as having been founded by Heracles (Ov. Met. 15.8ff,
Jans FL 50). According to Conon (above, n. 180), the Locrians left a place for Ajax in
tle is bate order; on Ajax as divine ally, see P. von der Mill, Der grosse Aias (Basel,
4} oth
ΒΔ] ἀαἀ ἩΠ er. Guthrie. Cf Hippoc. Morb. sacr, VI 258f L.
Todds, der. 145, after J. Bidez, Biographie d’Empédocle (Ghent, 1895). Heraclides
μη ν᾽ “Anvouy (frr. 76-89 Webrrli) takes as its starting point the last line quoted here
(Hh fet y).
IU. PYTHAGORAS IN THE EARLIEST TRADITION
extraordinary and amazing feats? His saga must have its roots not in
literature, but in reality; and indeed Gorgias testifies that “he himself
had been present when Empedocles performed feats of magic.""1™
According to Timaeus he restrained the destructive north winds at
Selinus by the use of bags made of asses’ hides—that is, by a secret
sacrifice in the manner of the Hyperboreans.1®*
A “journey into the underworld” was actually part of the ceremony
at the oracle of Trophonius in Lebadea.!8* The one who is to consult the
oracle was borne by a wind into the depths, feet first, as the dead are
carried. After the god has appeared to him, he returns to earth, often
after a lapse of several days. Before the sanctuary were the springs of
Lethe and of Mnemosyne, whom the initiate expected to see in the
underworld, according to the evidence of the Gold Plates. Trophonius
is a Ζεὺς XOdvios—Epimenides too met Zeus in the bowels of the carth,
But in Lebadea there can be no doubt that the main thing was ritual,
not legend. We do not know whether the visitor to the oracle was put
into a trance state—there were long ritual preliminaries, and not
everyone was admitted—or whether perhaps machines may have been
used, of the kind used in the katabasis of the Roman Bacchanalia,
according to Livy.!% It is significant, however, that legend connected
a Pythagorean, Parmiscus, with the otherworld journey of Lebadea.™
187 Satyrus ap. D.L. 8.59. Diels, SBBln 1884, 344, tries to weaken the force of chil
evidence by referring it not to a writing of Gorgias, but to a dialogue of Alcidaman:
cf. Burkert, RhM 1962, 48. |
188 EGrHist $66F30 = D.L. 8.60. One thinks of Aeolus’ leather bag. See R. Strémiberg,
Acta Univ. Gotoburg. 1950.3, 71-84, who cites additional comparative material. On the
Hyperboreans’ manner of sacrificing, see Pi. Pyth. 10.33.
189 The principal source is Paus. 9.39; cf. Hdt. 8.134, IG VII 3055, 4136, Rohde,
Psyche I 119ff = 92ff Eng. ed., Radke, RE VII A 678-695. Nilsson (II 450) does not believe
that the full development of the ritual came before imperial times. But Dicacarchus spoke
of a katabasis (frr. 1386), and said that “‘nescire ea melius esse quam scire” (fr. 17). Semon
(FGrHist 396F10) speaks of one who consulted the oracle and lost his ability to laugh.
Nilsson’s assertion that the “spring of Lethe” was an idea invented after Plato, and that
the spring of Mnemosyne belonged to the Hellenistic age (II 226f, Op. III 85~92) has been
refuted by the gold plate of Pharsalus, which has the phrase Μνημοσύνης Λίμνη (ca.
350 B.C.; Arch. eph. 1950-1951, 98ff). U. E. Paoli calls attention to the significance of the
fact that the visitor to the oracle is carried away feet first (Die Geschichte der Neaira | Bern,
1953] 43).
190 Livy 39.13.13: “raptos a dis homines dici, quos machinac illigatos ex conspectu in
abditos specus abripiant.” On this, see Festugitre, Mél. d’arch. et d’hist. 56 (1954) 94ff
(At p. 95, he conjectures that a mechanical contrivance may have been used in the Tro=
phonius ritual.)
191 Semos FGrHist 396F 10, and the Delian temple inventory of 279 Bc. (1 Χἧ 2.1611}}7,
DK 20). The age of the legend is indeterminable; the older inventory lists are all frage
mentary. It is possible that a dedicatory gift was ascribed later to Parmiscus. (fhe form
Παρμίσκος is found in the inscription, elsewhere often Παρμενίσκον, cf DK 20,). «
Apollonius of Tyana, too, was a visitor τὸ ‘Prophonius (Philostr, 1A 8.19).
3. Metempsychosis and “ Shamanism”
li many places there were subterranean installations which presented
the snderworld in physical form. The structures at Clarus are impres-
sive, the adyton with the spring that gives out the oracle lies under the
ella of the temple. And even more amazing are the subterranean
fussapeways at Baiac, near Cumae—if they really belong to the rites
af the “Cimmerians” and not merely to the water supply.1® There is
ala the mysterious mundus at Rome, visited by boys. In connection
with the cult of Demeter there were megara, subterranean rooms or
eaverns, into which offertory gifts were lowered.1%
This last brings us back to Pythagoras. His house, says Timaceus,
Was made into a temple of Demeter; and woe to the uninitiated who
entered τιΠ 96 In various segments of the tradition we have reports of
gulterrimcan chambers in which Pythagoras met with his disciples ;1%
and above all, we learn of a katabasis of Pythagoras himself, although the
glearest allusions to it are in the distortions of ridicule and parody.
Hieronymus of Rhodes said!** that Pythagoras descended to Hades
aid saw how the souls of Homer and Hesiod were atoning for what
Wty Wobert, Archacolog. Reports 1959-1960, 41f; “MunOévres eveBarevoar,” RPh 22
(ifyt) ayy. CL Tac. Ann. 2.54, above, n. 135.
Μὰ μι. Paget, In the Footsteps of Orpheus (London, 1967), thinks he has found the
μδυεείην mentioned by Ephorus FGrHist 70F134.
6 Cato ap. Festus p. 154 M.
δὲ ἡ γ]. Lucian p. 275.23 Rabe, Menander fr. 870 Korte, Paus. 1.27.3, 9.8.1, Plut.
fie fe ef Ov χγδς, Aclius Dion, s.v. μάγαρον, Paus. Att. s.v. μέγαρον, Hsch. s.v. Μέγαρα
Μ Seliede, Die Ruinen von Priene (19647) 93.—An inaccessible subterranean megaron
wae dinavercd a few years ago in Posidonia, where a potsherd was found with the
wide TM ΝΎΜΦΑΣ EMI HIA[PON (B. Neutsch, SBHeid 1957.2). Cf. also Hdt.
Δι:
Pettit 5606F131, lam. VP 143 (above, n. 128).
1 Antiphon Περὶ τῶν ἐν ἀρετῇ πρωτευσάντων (certainly not the Sophist Antiphon,
at Neath: tried to show in ZN [1.393 n. 3), Por. VP 9 = Iam. VP 27: In Samos, Pythagoras
ἐξιη tips πόλεως ἄντρον οἰκεῖον τῆς ἑαυτοῦ φιλοσοφίας ποιήσαντα ἐν τούτῳ τὰ πολλὰ
Re ἡμέρης καὶ τῆς νυκτὸς διατρίβειν συνόντα ὀλίγοις τῶν ἑταίρων. lamblichus adds,
μεμα ἦγ tom the same source, τὸν αὐτὸν τρόπον Μίνῳ τοῦ Διὸς υἱῷ διανοηθείς. Diogenes
Αμην ap Por. VP 34: ὁπότε θεῶν ἀδύτοις ἐγκαταδύσεσθαι μέλλοι καὶ ἐνταῦθα χρόνου
εν ἐλδιι ἐμίψειν, ἀλίμοις ἐχρῆτο καὶ ἀδίψοις τροφαῖς, like Epimenides (above, n. 174).
Hlipgodl Ref 12.18: ἐν ἀδύτοις καταγείοις ἠρεμεῖν ἐποίει μανθάνοντα .--- Τὸ this context
με μιν the subterranean basilica near the Porta Maggiore in Rome, on which see
Ε aber, Has, passim. To be sure, the interpretation of this as a cult site is controversial.
(Fob tvaatet, De dat van het grote hypogacum bij de Porta Maggiore te Rome [Diss. Leiden,
45} contea, CLC. van Essen, Muemosyne 4.13 [1960] 277-280.)
wei, ga Wo ODD. 8.21 (on the attribution of this to Heraclides, see above, ch. II 1
oa) Levy (Sources 37.1, Lég. 82) misunderstood the ironical conclusion and thought
the seferenee was to the punishment of adulterers (cf. Tam. VP 50); he therefore had to
atlieties the lant clause, καὶ δὴ καὶ διὰ τοῦτο τιμηθῆναι ὑπὸ τῶν ἐν Κρότωνι. Human life
ently Guly {ΠΠΠΠῸ by the consummation of marriage, and chose who remain ἀτελεῖς
will auiter a thought that is also expressed, in a different form, in the myth of the
Planabdes (Seo Carcopino, Bas, rat. 1, with references.)
II. PYTHAGORAS IN THE EARLIEST TRADITION
they had said about the gods (this does not sound very archaic),1**
and also how those were punished “who would not lie with their
wives; and this, of course, is why he was honored in Croton.” (The
gibe in this last clause is unmistakable.) Much more influential was the
account of Hermippus :?°°
After arriving in Italy, he built a little underground room and
instructed his mother to write down events, as they happened, on a
tablet, recording the time,?“ and to keep passing these notes down
to him until he came back up. His mother did this, and after some
time Pythagoras came up thin as a skeleton. He went into the
assembly and announced that he had just returned from Hades,
What is more, he read off to them an account of what had happened
during his absence. Taken in by his words, they wept and moaned
and were sure that he was some kind of divinity; so that they even Ὁ
entrusted their wives to him, thinking that they too would learn
something from him. And they were called Πυθαγορικαί.
The mocking tone of this account of a “(journey to the underworld”
is of course unmistakable. The question remains, however: how did this
sound when it was taken seriously; and how old is the story? The usual
assumption is that Hermippus’ account is a simple transference to
Pythagoras of what Herodotus and, after him, Hellanicus? had
reported of Zalmoxis, the Getic god; but it may be that the matter is
more complicated than that.
199 The background of this detail is to be found in Xenophanes and Plato, On the
Homeric poems in (late) Pythagorean tradition, see Delatte, Litt. 109ff, and M. Detienne,
Homéere, Hésiode et Pythagore (1962).
200D.L. 8.41, Tert. An. 28, Schol. Soph. El. 62 = Suda s.v. ἤδη. Allusions at Lucian
Gall. 18, Celsus (Origen C. Cels. 2.55), Eustathius on Od. 11.592, 24.264. See lDelatte,
Vie 244ff, Lévy, Sources 37ff. It is open to question whether there is a relationship te
Heraclides at this point (see ch. II 1, n. 32). The fact that Tertullian, the Sophocles scholiums
and the Suda connect this passage with Heraclides fr. 89, does not prove that Hermippu
himself had cited him (pace Corssen, RhM 1912, 23, and Lévy, Sources 40; Rohde, Psych
Il 419 = 599 Eng. ed., is hesitant).
201 τὰ γινόμενα εἰς δέλτον γράφειν σημειουμένην Kai τὸν χρόνον. Lévy, Sources 38.4
Lég. 1209ff, translates ‘“‘sealed letter,” and sees in the reading of ἃ sealed letter a furthe
miracle (cf. Philostr. VA 3.16, Lucian Alex. 19ff). But this leaves the words καὶ τὸν xpdve
unaccounted for; and one would expect σεσημειωμένην. σημειοῦσθαι means “mak
notes,” and the participle refers to τῇ μητρί. (The change of case is not uncommon, ¢
Eur. Med. 57f.)
202 Hellanicus, Βαρβαρικὰ νόμιμα, FGrHist 4F73. Here Zalmoxis promises the “return
of the dead and is therefore teaching a kind of metempsychosis, while Herodotus on!
speaks of immortality. The complete dependence of this book of Hellanicus upe
Herodotus (and Damastes of Sigeum) is stressed by Porphyry (Euseb. Pracp. evar
10.3 ΠΗ 41:72), but ic cannot be proven spurious (Jacoby 4.44).
3. Metempsychosis and “ Shamanism”
‘the Greeks called the Getae Γέτας τοὺς ἀθανατίζοντας, for this
wemcd particularly noteworthy to them. “They believe that they
ils not die, but that when someone succumbs he goes to the daimon
/aliroxis” (4.94). Every four years they send the god a “messenger,”
in the form of a human sacrifice. “But as I learn from the Greeks who
live on the Hellespont and the Black Sea, this Zalmoxis was a human
hein, a slave, in Samos, of Pythagoras the son of Mnesarchus” (4.95).
Zaliusoxis was sct free and returned to Thrace a rich man. There, thanks
te his Greek culture, he could easily impose on his fellows, especially
alee they were “a bit simple-minded”’ (ὑπαφρονέστεροϊ)), “Ῥεΐηρ
agyuainted with the Ionian way of living, and with manners more
polite than those of Thrace in that he had been familiar with Greeks,
ail with Pythagoras, who was not the meanest sage in Greece.”
Zalmwoxis built a banqueting hall in which he entertained the most
prominent citizens, and promised them that his guests and their
dewendants would not die, but would live forever in enjoyment of
everything good. In the meantime he had built an “underground
Bhamber” to which he now withdrew for three years, mourned as
deal. Vhen, in the fourth year, he reappeared; and now the Thracians
believed in him, and believed in the conquest of death and in immortal-
ity, “Concerning this underground dwelling,” says Herodotus, “I
ai neather excessively doubtful nor excessively credulous. I do think,
thuuph, thae this Zalmoxis lived many years before Pythagoras” (4.96).
It is not difficult to see the connection between the Thracians’
ἢ μι] and myth.2% Just as, once upon a time, Zalmoxis had disappeared
fur three years and reappeared in the fourth, so the god is summoned
every four years, by the “messenger”—summoned to a ritual meal,
whicl provides a guarantee of immortality.
Mile in Greek culture and Greek cleverness are obvious in the
δ 9}. the colonists gave Herodotus. They are far above Thracian
hathatisin; a slave among Greeks can be a god among Thracians and
make them believe anything he chooses.
Hemuippus’ story has been called a freche Uebertragung.? But the
8d thos sce BL Pfister, “ Zalmoxis,” Studies Presented to Ὁ. M. Robinson If (St. Louis,
just) bees 1121 (p. 1112 on the various forms of the name in Greek), There is now epi-
graplis al attestation οὔ Zalmodegikos” as the name of a Thracian prince (D. M. Pippidi,
μή εἰμ κι {τῶι} $3-66; SEG 18 [1962] no. 288).
*O Caen, RAM τωι, 43; similarly Rohde, Q 106 n. 1, Lévy, Sources 39, Lég. 133.
Rinee Heonuppus derived the doctrines of Pythagoras from “Jews and Thracians,”
{μεν dp 1.105), he doubtless knew Herodotus’ Zalmoxis story, and probably also
derepeeal lis dating of Zalmoxis before Pythagoras,
II. PYTHAGORAS IN THE EARLIEST TRADITION
Greeks were often inclined to attribute to foreign peoples things that
were really Greek, and Herodotus in particular mentions many things
in his accounts of foreign peoples which he intentionally omits when
writing about Greeks. The most striking example is the Egyptians’
alleged doctrine of metempsychosis, but the situation seems to be the
same with abstinence from beans? and with the novella about
Rhampsinitus’ treasure house.2° Herodotus tells about werewolves
among the Neuroi of Scythia, but has nothing to say about similar
phenomena in Arcadia.?®
The striking thing about the Getae is their belief in immortality,
and this must have been the reason Herodotus’ informants thought of
Pythagoras. Ionian manners and Ionian cunning could have given
Zalmoxis ideas in other realms, but Pythagoras was the authority in
questions of the afterlife and of immortality,?°* as we learn not only
from Ion but from Herodotus himself. This reputation must have been
familiar to the Greeks in plain and memorable terms. Herodotus
assumes that one knows what he means when he says Pythagoras was
“not the weakest” σοφιστής of the Greeks. Therefore, it will be well
to consider whether specific details of the Pythagoras tradition are
reflected in the Zalmoxis story.?°*
It is doubtful whether the “subterranean chamber” really belongs
in the Zalmoxis tradition. According to a report from the age of
Caesar, there stood beside the king of Thrace a revered pricst of
shaman who was a successor of Zalmoxis—at that time a certain
Decaeneus, beside King Burebistas—and he dwelt, as Zalmoxis did,
on the holy mountain Cogaconus.”!° Given the strong tendency for
205 Above, n. 48.
206 Hdt. 2.121. The cult legend of Trophonius and Agamedes is very like it, while on
the other hand no Egyptian reference to it has been found (cf. Radke, RE VII A 680, Lévy,
Lég. 187). In the Rhampsinitus story as with Trophonius, a katabasis is involved. According
to one version (Schol. Ar. Nub. 508), Trophonius was starved in a subterrancan chamber,
as Pythagoras, too, died of starvation (Satyrus ap. D.L.8.40, Dicaearchus fr. 35).
207 Hdt. 4.105—Plin. HN 8.81, Varro ap. Aug. De civ. D. 18.17, Pl. Rep. $6sd.
208 After Zeller (SBBIn 1889, 992), Rathmann reluctantly draws this conclusion fron
Hat. 4.94ff. So does Maddalena (347ff), though he tries also to find that this is in contradies
tion to the doctrine of metempsychosis. Cf, however, the version of Hellanicus, un. 204
above. On Ion fr. 4, and Hdt. 2.81, sce above, nn. 13, 39.
209 Boyancé, Muses 134, finds here the most ancient reference to the special significance
of cult meals for the Pythagoreans. Morrison, CQ 19$6, 1374 goes further, trying to
find a social and political significance: in the ἀπάγοι Pythagoras presents his doctrines to
the assembled lonian bourgeois, a σοφιστής like Xenophanes, But it scems that the
essential point is in the following trick, without which not even the ‘Thracians would
believe in immortality.
0 Strabo 7.2970.
3. Metempsychosis and “ Shamanism”?
teliprous traditions to cling to holy places, we may believe that in
Herodotus’ time, too, the Thracians thought of Zalmoxis as being
on his holy mountain, and not in an underground dwelling somewhere.
Hint sf this is so, this is a Greek motif; and it may have been that the
(iteeks imputed to Zalmoxis rather a slavish imitation of Pythagoras.
Ilermippus’ account surely cannot, in all respects, be derived from
Herodotus. The notes passed down into the underground room could
he explained as an elaboration of the trick; Pythagoras must know who
has lied in order to describe his experiences in Hades in a credible way.
Hit how does it happen that his mother is his confidante and assistant?
lt is hiphly unlikely that Pythagoras brought his mother with him to
Croton, and such an idea is never mentioned in the tradition. What
we hve, then, is a rationalizing version of something quite different.
Pythaporas brings with him from Hades τῆς μητρὸς παραγγέλματα
(vommands of “the mother’), a message from the divine Myjrnp—
Peneter2"! Thus the “‘little dwelling” becomes a sanctuary of Deme-
fer, as ‘Timacus says Pythagoras’ house was (above, n. 196). In this case,
huwever, Ilermippus has an element of the story that does not derive
frum) [lerodotus, and whose significance is no longer understood—
therefore something quite ancient, and belonging originally to the
Pythaporas legend. Another feature of the story that makes the
linpression of being genuine and ancient is the wasting away of
Pythayoras. Intensive fasting always forms a part of the routine of
alianians and fakirs.2!2 This is not mentioned in Herodotus’ account
elthier 2" so that Hermippus’ report has independent value as evidence
alonpside that of Herodotus. It shows Pythagoras in the role of a
Werophant in the cult of Demeter.?™4
Vhe most remarkable detail of the Pythagoras legend, his golden
thigh, points in the same direction. Antiquity understood this as a
sign ol divinity, but we find no explanation of just how this is so.?¥°
“004 the intentionally ambiguous characterization of Callias as ἱερεὺς τῆς μητρὸς Kai
εῆν thyarpay in Andoc. De myst. 124 (L. Koenen, Studien zur Textgeschichte und Text-
beitik [C Glopne, 1960} 87.)
ἩΣῈ 4 the ἄλιμον of Epimenides (above, n. 174).
#4 A oun your fast is impossible, as Herodotus knew. But the four-year cycle was an
rasetial feature of the Thracian ritual (above, n. 203); so this detail had to be dropped
free the: Zalmoxis story.
0 t Ach fr. τὸ, on a megaron that may only be entered by hicrophants.
ἘΠῚ, was obviously considered a purported proof of divinity as early as Aristotle
(lau Ef ago, above, no 110). A scholium on Lucian (p. 124.6f Rabe) tells us that the thigh
ab Pythaporas was imprinted with an image of Apollo (ἐντετυπωμένον), Cf Dodds’s cau-
fits 1 Κὰ fr. 163 0. 43. Lévy, Sources 12, Leg. 4g nt, interprets both the golden chigh
amd the cnage of Apollo as phenomena of the sun's glare; but why precisely the thigh?
1. PYTHAGORAS IN THE EARLIEST TRADITION
Long ago Wilhelm Mannhardt made reference to myths of dismeme
berment and revival;##* but here again, the legend is based on the
actuality of the initiation rite. It was merely imitation of Pythagoras
when Alexander of Abonuteichus, at a mystery ritual, let a glimpse of
his golden thigh be seen, but it is worth noting that this festival itself
was organized on the pattern of the Eleusinian.24”7 More important is
what Prudentius says in our most detailed description of the cult of
the Mother, namely that persons dedicate themselves to the Mother
and receive her “seal” (sphragitis), which is burned into them with rede
hot needles; and at the burial of an initiate the “dedicated member”
of the body was covered with a gold plate.2!8 Tattooing of the Galli is
attested for the Hellenistic period.*1® But, before all, the myths tell
over and over of the favorite of the Great Mother being wounded in
the thigh,?° as also of the thigh wounds of those who attempt to make
their way into the underworld.?#! Only he who bears the sign can
descend into the pit with impunity.22* In the same way, Pythagoras’
golden thigh is the sign of the initiation which makes it possible for
him to travel to Hades.
Are we to suppose, then, that Hermippus’ whole story, including the
golden thigh, is literally true? In any case it contains reflections of
ancient ritual practices which survived for a very long time.? And
though there is no direct testimony on the matter before Aristotle,
Hieronymus, and Hermippus, we must take account of certain possible
allusions in earlier literature. In the Electra, for example, Sophocles
216 Germanische Mythen (Berlin, 1858) 74. Against Mannhardt, Meuli (16o0ff) cites
Siberian myths of “golden heroes” and “ golden gods.”—On dismemberment and revival
in shaman initiations, see Eliade, 45ff, 53.
217 Lucian, Alex. 38, 40.
218 Prudent. Perist. 10.1076ff; F. J. Dolger, Sphragis (1911) 41ff.
219 Etym. magn. s.v. Γάλλος, Plut. De adul. et am. 56e, 3 Macc. 2.29.
220 Adonis; for Atys (or Attis), see Hdt. 1.34-45.
221 Heracles bitten by the serpent in the tail of Cerberus, Apollod. 2.5.12; cf., for
example, a volute crater in Munich, no. 3297 (Furtwaingler-Reichhold pl. 10).—On
Theseus, Schol. Ar. Eq. 1368; on the death of Miltiades, Hdt. 6.134. Cf. Phronesis 14
(1969) 22-27.
222 The Galli in Hierapolis, Strabo 13.629, Damasc. Isid. 131; Dionysus: Hor. Carm,
2.19.32.—The Ethiopians tattooed their children on the knee with an image of Apollo,
according to Lydus Mens. 4.53 p. 110 Wuensch.-——Each of the Seleucid kings, descendants
of Apollo, had a birthmark on his thigh (Justin 15.4.3-5).
223 There are surprisingly similar stories of the prophet living underground and sub.
sequently appearing in Franian-Arabic tradition (G. Widengren, frarnisch-semitische
Kulturbegegnung [960] 62-66).
3. Metempsychosis and “ Shamanism”
las (Orestes say that he will not scruple to give a false report of his
death 4
Why yap εἶδον πολλάκις Kai τοὺς σοφοὺς
4 4 , - > ᾧ ’
λόγῳ μάτην θνήσκοντας" εἶθ᾽, ὅταν δόμους
ἔλθωσιν αὖθις, ἐκτετίμηνται πλέον.
The wise” win special honor by being regarded as dead for a time and
then returning. The stories of Aristeas and Zalmoxis, as told by Hero-
ditis," provide excellent examples of this; but would it be these
persons that would most readily come to an Athenian spectator’s
fitnd, or rather Pythagoras—EAAjvev οὐχ ὁ ἀσθενέστατος σοφιστής
(Hilt, 4.95)? He was, after all, Zalmoxis’ teacher; and it is to him that
the ancient scholia refer.
ΠΟ ας. calls Pythagoras the chief of swindlers,?26 and accuses
Wim of having made, from the books of others, his own σοφίη, πολυμαθίη,
ΝΗκοταχυίη 2327 Commentators have emphasized the words σοφίη and
φολυμαθίη, and paid much less attention to the climactic word κακοτεχνίη.
Wye is not a doctrine, but an activity; κακοτεχνία is a technical term
fir the subornation of perjury, and in general designates disingenuous
fies by which anyone attains an end.228 Thus Heraclitus is accusing
Pythaporas of being a charlatan who, by ignoble deception, has
Attained to fame. The kind of procedure that Herodotus attributes to
Faluwoxis, Hermippus to Pythagoras, and that the story of Aristeas in
Metapontum presupposes, is unquestionably a glaring example of
RHnureyedy; and it seems quite likely that Heraclitus, as well as Herodotus
atl Sophocles, had heard of a ritually enacted katabasis of Pythagoras.
MO iyl EL 62ff. To be pronounced dead was to the primitive mind a serious matter,
#8 event scarcely to be undone (cf. Plut. Quaest. Rom. 5).
6. Kabel refers to Zalmoxis (Sophokles Elektra (Leipzig, 1896] 79). On the relation-
ship vt Herodotus and Sophocles, see Jacoby, RE Supp. II 234ff. Heracles, Theseus, and
ι it yeu ac arccely are relevant cases here, since, though they were thought for a time to be
Heal aul gone, this was involuntary, and not the result of their σοφία.
"νη ίδων ἀρχηγός, fr. 81 (Timaeus FGrHist 566F132). Cf. Wilamowitz, Hermes 62
{iuse) ayy, Reinhardt, Hermes 63 (1928) 107-110.
FP tay; above, n. 63.
“θα, PCP 134.2, Reinhardt, Parm. 235f. Ciaceri II 94. Hera’s ἀπάτη (Il. 15.14) is
wundieqens., so is the corruption of the Pythian priestess (Hdt. 6.74), the introduction of
lyitie withesses (Dem. 47.1, 49.56, Pl. Leg. 936d), false artifice used by a musician (Ath.
bagel) Mhetoric is κακοτεχνία to Epicurus (fr. 51 Us. = 19.3 Arrighetti).—Cf. also
(πεν 000474), in the context of ἀγυρτικόν and γοητεία the expression τὸ φιλότεχνον
εἰ μεμὶ εἰς “Πιονυσιακὰς τέχνας καὶ τὰς ᾿᾽Ορφικάς. M. Marcovich, Philologus 108 (1964) 41,
θὰ μευδέων τέκτονας καὶ μάρτυρας (fr. 28); the threat uttered in this fragment
μα tee dns cine of fr. 66 and 14, and we are in the realm of the μύσται.
II. PYTHAGORAS IN THE EARLIEST TRADITION
Werner Jaeger wrote of Pythagoras, “The modern fashion of describe
ing him asa sort of medicine-man has no claim to serious consideration.”
But an examination of the most ancient evidence makes it difficult to
forget about the wonder-worker. The concept that first made it
possible to take this aspect of the tradition seriously was that of shamans
ism, introduced by Meuli and Dodds.”?®
The word “shaman” comes from the language of the Tunguses of
Siberia, and the phenomenon of “‘shamanism’”’ was first studied in
relation to certain Siberian tribes. The shaman has the ability, in an
ecstatic state which is voluntarily induced by means of a definite
technique, to make contact with gods and spirits, and in particular to
travel to the Beyond, to heaven or to the underworld. Shamanism is
the focus of these peoples’ religious and intellectual life; the shaman’s
special task is to bring back health for the sick from another world
and to conduct the souls of the dead to their new home. There are
female as well as male shamans. The ability to achieve the shaman’s
ecstasy, and the technique necessary, are won, by those who have a
special vocation, by a long ceremony of initiation, which includes the
imparting of a certain mythical “knowledge,” comprehending earth,
heaven, and underworld.
The ecstatic journeys of a Hermotimus, and the ecstatic healings of a
Phormio or Leonymusare obviously analogous to this, and the apparent
death and the journey of Aristeas seem easier to understand, like the
“flying”? of Abaris on his arrow. The connection of Aristeas and
Abaris with central Asia is certain.22° Dodds speaks of the Grecks as
coming into contact with shamanistic ideas from the north at the period
of colonization of the Black Sea area; and he also brings Pythagoras
into this context. Here we find, all together, association with gods and
spirits, mastery of animals, disappearance, and simultancous presence in
different places.23! To be sure, the katabasis of Pythagoras is not of a
228 Jaeger, Paideia 1 162 tr. Highet (= I 221 Ger. ed.).—Meuli and Dodds first worked
this out in detail, but Rohde had already referred on occasion to Eskimos and Indians
(Psyche Il 97.1), and Diels had referred to shamanism (AGP 1897, 233ff, Parm. 14f, ΝΙΝ
1922, 229) as Lobeck, I 13 ἢ. h, had already done.
230 The Arimaspi have their origin in central Asiatic mythology (A. Alféldi, Guomon
9 [1933] 566ff, Meuli rssff). It has been recognized since the work of Meuli that
the Scythians and Thracians had a genuine shamanism (Eliade 376ff); cf. the Thracian
καπνοβάται (Posidonius PGrHlist 87F104 Strabo 7.296) with the al@poBdrns Abarin
(above, n. 162).
231 Above, nn. 117-125, Eliade 103ff Apollonius of Tyana also learned the language of
animals (Philostr. VA 1.20). It is a common motif that an eagle brought the power of
shamanisin from heaven (Eliade 78}.
3. Metempsychosis and “ Shamanism”
apecilically shamanistic kind, any more than the miracles credited to
Kinpedocles, Epimenides’ visits to the cave, or the visit to the under-
world from Lebadea.
‘Whe significance of the idea of shamanism for the history of philoso-
pliy lies in the conjecture that the new conception of the soul, which
waa to become the dominant one through the influence of Plato, is to
he traced to this source. The independence of the soul from the body
fa linmediately experienced and depicted in the shaman’s ecstasy; and
this is the reason for the particular interest taken in these phenomena
by the Platonist Heraclides and the Peripatetic Clearchus (frr. 7-8).
The ‘Vhracian Orpheus is not far from Scythia,?8? and in this vicinity
(geographical and religious) we are but a step away from the doctrine
af imetempsychosis. The belief that the spirit of a powerful revenant
has entered a living body is common among shamans,” and precisely
this is attested for Epimenides. The inspired bard feels himself at one
with the whole world: “Thave been an eagle, Ihave been a sea coracle...
Ε have been a sword in the hand, I have been a shield in battle, I have
been a string in a harp.’2% It only needs a small stimulus, though its
Geequences are important, to make such ideas into a full-blown theory
ΘΓ netempsychosis.
The fact is that the very richness of the Greek tradition in stories
af this type is somewhat embarrassing. At least some of them go back
farlier than the colonization of the Black Sea. Meuli attempted to
derive Greck epic from shamanistic poetry, and Odysseus does have
Hhanvanistic traits.22> Then we must add Melampus’ learning the lan-
ΓΝ f the animals,?* Tciresias’ changes of sex,?*” Polyidus’ awakening
8 Cy ΟΣ as a shaman see Dodds, Irr. 147ff.
ΜῈ Yivulils, fre. 144f, Eliade g1ff. Shamans also claim to be descended from the sky-god
(ΕΠ μεν pot). Kalin emphasizes (AGP 1960, 32ff) that a doctrine of the nature of the soul
dies tot necessarily develop from such ideas.
#6 fayvesuons of Irish bards, cited by Cornford (PrSap 122). “It is difficult,” he says,
* ti elecide whether transmigration or metamorphosis is meant ... But the difference
fie} Ween metamorphosis and metempsychosis is, after all, not great” (123). Cf. Empedocles
fi ii7, “lon Ehave been ere now both boy and girl, and bush, and bird, and mute fish
iis tla waves” (tr, after Leonard), Dodds derives Indian doctrines of metempsychosis
ἤει ehatuanisn (rr. 172 ἢ. 97, citing Acta orientalia 17 [1939] 164ff).
me Mele ΠῚ ἡ, Schwartz (Die Odyssee (Munich, 1924] 1858) called attention to
“ghansaniatie features of the story of Odysscus—aside from his narrative of his journeys
iv He fit person and his journey to the land of the dead, also his assumption of the role
vf a πέμμην, and especially his activity as an ἀγύρτης in Thesprotia.
Mile th 261 M.-W., Apollod. Bibl 1.9.11. Shamanistic parallels in Eliade τὸς
ἢ Fits
lew tr 274 ΜΙ. (the Melampodia), Parallels from shamanism, Eliade 248.
II. PYTHAGORAS IN THE EARLIEST TRADITION
Glaucus from the dead,?88 the motif of the winged seer,?9® and the
magic quality of music, which can be ἐπῳδή, an incantation with
power over gods and spirits.24° Then there is the Messenian legend of
the magician and king Periclymenus, who could change himself into
any sort of creature,™4! and of Aristomenes, who made his way back |
out of the chasm of Ceadas.?42 In addition, if γόης, a word that come
bines the magic of self-transformation with the mourning of the dead,™@:
originally meant something like “‘shaman,” this takes us very far back
indeed.
It is a controversial question, however, how far we may go if
assuming a general spread of the kind of shamanism found among the
peoples of Siberia and among the Indians, and how this phenomenon
is to be fitted in to the general history of mankind.” It is especially
difficult to be certain about the existence of an Indo-Iranian, or @
specifically Iranian, shamanism.**° The mythical motifs concerned catt
be traced with certainty back to the early ages of Mesopotamiaf,
civilization—Innana’s trip to the underworld, Gilgamesh’s search for
immortality along the path of the sun beyond the great twin-peaked}
mountain.“ The pertinent ritual, the cult of the Great Goddess,
probably goes beyond the Neolithic to the Paleolithic Age, and we may
238 References, RE XXI 1653ff. Aeschylus knows the legend, and the well-know#
fragment 638 of Euripides comes from his Polyidus. Furtwangler, Gemmen III 253, identified
representations of the legend on archaic Etruscan gems. See also R. F. Willetts, Klio 99
(1959) 21ff. The Sotades cup (London Ds) shows Polyidus and Glaucus in a subterrancag
vaulted chamber labeled “Hades” (A. S. Murray and A. H. Smith, White Athenian
Vases in the British Museum [London, 1896], pl. XVI; J. Beazley, Attic Red-Figured Vases ,
Painters? [Oxford, 1963] 763 no. 2; ca. 450 B.C.). .
239 Musaeus has the gift of flight (Paus. 1.22.7) See further P. Wolters, “Der gefliigelee
Seher,”” SBM 1928, 1. οἵ
240 Cf. Boyancé, Muses (though he avoids the word “shamanism”’).
241 Hes. fr. 33 M.-W.
242 Paus. 4.18.5: An eagle, or the air resistance of the shield, bears Aristomenes ut
harmed to the floor of the ravine, just as, at the Trophonius oracle, a wind bears the
inquirer into the depths.—The Dioscuri take Aristomenes’ shield from him, and he
finds it again in the sanctuary of Trophonius (Paus. 4.16.5-7). A 4th-century tradition
connects Aristomenes with the mysteries of Andania (Paus. 4.26.8). Interestingly enough, -
there are close connections between Messenia and Magna Graecia; the colonists of
Metapontum came mainly from Messenia (Kicchle 34ff). Ganschinictz lists more katabaglg
stories from Greece, RE X 2395ff. There are also relationships with Italian material: the
Ausonians related of Mares, the “first man,” whose form was that of a centaur, that he
lived 123 years, and τρὶς ἀποθανὼν ἀνεβίω τρίς (Acl. ΜῊ 9.16).
243 Cf. Burkert, RAM 1962.
244 Eliade regards shamanism in a quite general way as an ecstasy-inducing technique,
L. Vajda writes against this generalization (Ural-Altaische Jahrbilcher 31 [τῷ 40] 456-484,
with full bibliography); he would like to narrow it down as closely as possible-=
geographically, phenomenologically, and historically.
860 CFG. Widengren, Namen 2 (1044) 6267.
M6 ANE 1 62-$7, 47-90, 88-96; Phronesis 14 (196y) 16-21.
3. Metempsychosis and “ Shamanism”
samjecture that from very early there were attached to it societies of
tient with their initiation ceremonies, bringing the renewal of life,
possibly in wild, ecstatic orgies. The sloughing of a snake’s skin,
attributed by the Gilgamesh epic to its having swallowed the plant
af immortality, is described in an early Greek poem in the words,
"Only the ψυχή remains.”*47 This complete separation of body and
tint! docs not seem, however, to belong to the early Mesopotamian or
vatly Mediterranean tradition, whereas in the Iranian tradition the
dichotomy of “bony” (that is, corporeal) and “spiritual” life is firmly
eatablished.248 When Aelian® tells us that Pythagoras wore white
glothing, a golden crown, and trousers, this last detail is a sure pointer
foward che Iranian-Scythian area.25° And yet these northern nomads
Hever developed a doctrine of metempsychosis, so that we are once
flute tempted to consider the possibility of a direct dependence of the
Greeks on the Indians (above, n. 71).
It ix not possible, at this late date, to search out and map every
Ehannel of historical influence. Whether or not the expanded concep-
Hon of shamanism is recognized as legitimate is a terminological
Weation for specialists; but it has in any case performed the useful
Ai tion of taking the so-called myths and legends seriously and show-
tag how they make sense as clues to actual cult practices.
Fin this is the picture of Pythagoras that emerges from the study
af the most ancient testimony, not influenced by Plato. He is the
hletuphane of Great Mother mysteries with an Anatolian stamp, and
has as new doctrine, probably influenced by Indo-Iranian sources, of
Immortality and of the triumph over death through successive re-
births. Epimenides and Empedocles were similar “shamans,” but the
apecial character of Pythagoras’ activity is seen above all in one fact—
" ἐπ πο existence of a society of Πυθαγόρειοι.
8 Pew fi. 204, line 139. M.-W.
ΜῈ Avesta, passim. The teaching of the magi on resurrection and immortality:
PWereqeomipus ΠΗ ΙΕ 115F64 = D.L. 1.9.
M1 lis 32: Πυθαγόρας ὁ Σάμιος λευκὴν ἐσθῆτα ἤσθητο καὶ ἐφόρει στέφανον χρυσοῦν
eal ἠνηξυρίδαν, In exactly the same αἴεγε---έν χιτῶνι λευκῷ καὶ χρυσῷ στεφάνῳ καὶ
deafipiss the highest God, lord of death and rebirth, appears in the “‘ Mithras liturgy.”
at fap owe mag. IV 699 = A. Dieterich, Eine Mithrasliturgic (19235) 14, 15 (a reference
Hipplicdl ἀν τ by M. West). Cf. the white clothing of the Pythagoreans, attested at
1 400, lain. VP 100 (probably from Aristoxenus, since Aristox. fr. 27 occurs in both
μεν Mastagni, Verbo 272.1, Boyancé, REG 1939). Also note the golden crown in the
᾿νε at Archytas” at Tarentum (AA 1927, 137; Glotz—Cohen, Hist. Gr. Il [Paris,
dah} ges 70; Wuilleumier $48f), The Messenians, too, buried their dead with golden
ΕΗ ΨΧῊ andl white clothes (Paus. 4.17.2f).
ect bldt. 1.71--—-Pythagoras is represented with a turban, as in Καὶ. Schefold, Die
Bibditeer der antiken Dichter, Reduer wid Denker (Havel, 1943), τοσοῦ, 160.1.
I, PYTHAGORAS IN THE EARLIEST TRADITION
4. ACUSMATA
The oldest form of transmission of the teachings of Pythagoras ig
represented by the acusmata, which are also called symbola, orally
transmitted maxims and sayings.1 Our first tangible evidence about
them goes back to about 400 B.c. Anaximander of Miletus (the:
younger), whom Xenophon names as one of those who could find’
the hidden meanings in Homer, also wrote an Explanation of Pythag
orean Symbola.* This shows that the tradition was pre-Platonic; and ἡ
these symbola already needed explanation, or allegorical interpretation
like the text of Homer, they must be much older, must go back in fa¢
to pre-classical or archaic times.
It is a matter of question how much of the detail we have goes bael
as far as Anaximander’s book.’ Once again we find our most importan
evidence in Aristotle’s book on the Pythagoreans. In addition t¢
fragments cited by name,‘ there is a long passage in Iamblichus which”
1 Holk, a student of Rohde, first clarified the main lines of the tradition. Later, Boehm
collected comparative material from folklore, which helps to interpret the various symbelgg
In the following pages his numbers are given, and these should be understood alsa ag
referring to his convenient collection of parallel references. The most extensive discuslom
is that of Delatte, Litt. 269ff; the most important texts are in DK 58C.
2 ἔγραψε Συμβόλων Πυθαγορείων ἐξήγησιν (Suda s.v. Anaximander = FGrHist 9 Fy
= DK 58C6). Corssen, RhM 1912, 249f, followed by Lévy, Sources 67.3, maintained
that the Suda confuses Anaximander of Miletus with Alexander (Polyhistor) of Milettt,:
who also wrote on the Pythagorean symbola (FGrHist 273F94 = Clem. Al. Strom. τιθῇ
Jacoby, in his note on the passage, disagrees with Corssen). Xenophon, however (Symp
3.6), refers to Anaximander’s method of allegorical interpretation, and we cannot hop
for any older or more apposite confirmation. The Suda dates Anaximander in the τομῇ
of Artaxerxes Mnemon (405-359). The dramatic date of the Symposium (422 B.c.) prover
nothing, as Xenophon even introduces himself as a character (1.1; cf. Ath. 5.216d), Iw
the mention of Stesimbrotus together with Anaximander brings the latter closer τῷ 46
than say, to 360.
3 The Suda (s.v. Anaximander) cites three symbola: not to step over a yoke, not to ath
the fire with a knife, and not to eat from a whole loaf. The first two are favorite example
of Pythagorean symbola (no. 30 and 33 in Boehm; cf. Por. VP 42), so that one is led t
suspect that the Suda’s datum is not taken from Anaximander’s book but, for example
from some handbook. On the other hand, the third example (no. 39 Bochm) is na
found elsewhere except in Hippol. Ref. 6.27.5, where the other two are also found (wlth
σάρον instead of ζυγόν). Delatte (Lift. 286.2) supposes that Hippolytus and the Suda are
both dependent on Anaximander.
* Cf. H6élk 21-40. The group of fragments collected by Rose (194-197) and printed
without change by Ross must be revised: fr. 197 does not belong; Jerome copied Pots
phyry carelessly, and the name of Aristotle (Por. VP 41 Arist. fr. 196) got attached τὸ @ |
passage which Porphyry expressly distinguishes from the Aristotle citation (Rohde, Q
139.1, HOlk 186, Por, VP 42 Jerome belongs in the Androcydes eradition).—Fr, 19g
is arbitrarily singled out by Rohde; D1. 8.33 belongs to the Memoirs section (HOlk a4
36ff, DK «Β|}1τ, ΠῚ 27383). Pr. ya, firse part (lam. {ΠΡ 20) is not an excerph
but Tamblichus’ own composition.
4. Acusmata
hae preserved Aristotelian material, partly in his very words.® Philo-
Haris also wrote a book Περὶ συμβόλων. But the principal source of
the later tradition is the book of Androcydes, a Pythagorean, Περὶ
[hiyopucdv ovpBoAwy.? A physician named Androcydes is cited in
fiauth century books;® but there is some literary fiction involved in
thie context,” and it is not clear just what the relation may be between
this and the book on the Pythagorean symbola. In any case the latter
Waa in existence in the first century B.c.1°
lw the above-mentioned passage of Iamblichus, the acusmata are
divided into three groups and given as answers to the questions τί
ber, tf μάλιστα; and τί πρακτέον; In the first group there are two
:
i δ, 1 82-86, according to Hélk’s analysis (3 1ff) pp. 47.11-50.17 Deubner. But one
; Way well wonder whether p. 47.4-11 does not also have Aristotelian material, especially
ee the passage is connected closely with Ael. 4.17: πειρῶνται διαφυλάττειν ὡς θεῖα
ypara (lain.) ~ ws χρησμῷ θείῳ προσεῖχον (Ael.); ἀκούσματα ἀναπόδεικτα καὶ ἄνευ
ἤνην (lai ) ~ οὐχ οἷόν τε δὲ ἦν διαπορῆσαι ὑπέρ τινος αὐτῷ ἢ... προσερωτῆσαι (Ael.). The
δὲ ΜΝ δ νον in lam., τούτους ἔχειν βέλτιστα πρὸς φρόνησιν οἵτινες πλεῖστα ἀκούσματα
Saxo, baa μ΄ Aristotelian ring, too.
Pt eddie 2811, doubtless with reference to Pythagoreanism (Jacoby IIIb Supp. 228f).
HOt ΗΘΚ ΔΓ The oldest citation is by the rhetorician Tryphon RhGr III 193f Spengel.
Vey clewely connected with this is Plut. De educ. puer. 17.12d-e, as is confirmed by the
PMiatlene at Androcydes in Paroemiogr. gr. II 437, 770. Similar is Demetrius of Byzantium
δ. Ath to.4s2d-e; and other passages to be included here are D.L. 8.17, Por. VP 42,
Mbp Ner/. 6.26-27 (sce HOlk’s tables, soff). It is not known, though, whether the work
8 oe Volyhistor or Anaximander of Miletus contributed importantly to the
ee along with Androcydes. One trace of Ionic dialect (μαχαίρῃ) is found as late as
barely (100 Us. et Os. 3546.)
δ Pheuplu. fist, pl. 4.16.6 (whence Plin. HN 17.239), a formula for the prevention of
jnkeniens; Clearchus ap. Ath. 6.258b, on the flatterer (Wehrli breaks off fr. 20 before
is paunye, though che sentence following certainly refers to 255d = Clearchus fr.
fo W nde ὑπυμένειν ~ πάντα ὑποδύεται; ταπεινὰ ἤθη ~ εὐχέρεια; καταφρονητικοὶ τῶν
δ nina, ~ οὐ βαρυνόμενος οὐδενὶ τῶν αἰσχρῶν. Kaibel has the limits right in his
“aid
junit Ath). Theophrastus is probably dependent on the letter of ‘““Androcydes, well
Hwee fin dius wisdom” to Alexander the Great, whose ‘“‘lack of moderation” he wished
fe “iedteain” (Plin. HN 14.58). There is a citation containing a warning against over-
μμήπίμεηι rin wine and meat at Clem. Al. Strom. 7.33.7, and, without the author’s name
bu iw dente dialect, ae Plut. De esu carn. 995e; cf. De frang. 472b.
# Ube sli attatic situation of the “letter to Alexander” is surely imaginary, and the person
ff the autho may be so as well. Corssen believed in its authenticity (RAM 1912, 244ff),
bit fie iain argument, supposed Pythagorean dietary regulations in Androcydes, is a
Hiisuilertancding of Plin. AN. 17.239. He was followed by Delatte, Litt. 285, Vie 186,
δ bey Wertermann (passim), for whom Androcydes becomes one of the main sources
afithe wlhode Pythagoras tradition. Diels (DK I 465 n.) and Lévy (Sources 66ff) express
debts, and a negative judgment is given by Zeller (III 2.118 n.) and Hélk (4off).
Anson ydes coincides a number of times with the ‘Three Books,” a forgery of
Ghent μα ας attributed to Pythagoras (cf Burkert, Philelogus 1961; D.L. 8.7, οὗ, 14).
Far example, one ought to pray, not for what he wants, but for what the gods want
(HY πον ch Diod. 10.9.7.—"‘Androcydes” in Tam. VP 145, in the framework of an
dtierdote about the Pythagorean Thymaridas), and the calculation of the length ofa period
Hf pel εἴθ (DLL. 8.14; Androcydes ap. ‘Ph. ar. 52.8; ef. ch. I 4 on. 110). Here the
lity is not ascertainable with certainty; ef mn. 74; a sentence on the quadrivium, in
dislect, occurs in Nicom. Ar. 1.1.1.
1. PYTHAGORAS IN THE EARLIEST TRADITION
instances of verbal agreement with Diogenes Laertius’ citation from
Aristotle" and a third passage agrees almost exactly with one in the:
Oeconomica attributed to Aristotle. Here a textual corruption in the!
latter may be removed by use of the Iamblichus text.!? Some sentences
in Aelian, quoted along with other material about Pythagoras,#
have points of contact both with fragments of Aristotle and with
lamblichus. The conclusion seems inevitable that not only the passage
of Iamblichus"* but also that in Aelian go back to Aristotle.
The other two categories of acusmata in Iamblichus can also be
traced to Aristotle. Porphyry cites, from Aristotle, some sentenced
of the same type as the τί ἔστι adages in Iamblichus.’® The pronounces
ment that Pythagoras was “the Hyperborean Apollo”’ is expressly
attested as an acusma;* in the passage just mentioned, Aelian has som@
material that is closely related to what is attested by Porphyry;
there is similar material in Diogenes Laertius,!* and also a remark
Aristotle himself.19
11 Jam. VP 84, p. 49.6 D.; D.L. 8.34 (Arist. fr. 195; Diogenes’ version is more detailed)
Iam. VP 86, p. 50.8 D.; D.L. 8.35 (ἄρτον μὴ καταγνύειν). The last of the explanatle
given by D.L. (ἐπεὶ ἀπὸ τούτου ἄρχεται τὸ ὅλον) must go with the sentence in Iamblich
od δεῖ οἰωνὸν ποιεῖσθαι τοιοῦτον ἀρχόμενον καταγνύντα καὶ συντρίβοντα, whether
cosmological interpretation proposed by Delatte (Vie 239) is wrong, or whether lata
blichus misunderstood. 4
12 Tam. VP 84, Ρ. 49.4 D.: γυναῖκα οὐ δεῖ διώκειν τὴν αὑτοῦ; ps.-Arist. Occ. ἀρνὶ
(DK 58C5): ἥκιστα δεῖν δοκεῖν ἀδικεῖν. Instead of the usual reading δεῖν [δοκεῖν] ἀδικῇ
(as in DK) should be read δεῖν διώκειν [ἀδικεῖν] (Deubner, SBBin 1935, 672). Cf. alia
Iam. VP 48.
13 Ael. 4.17, καρδίας ἀπέχεσθαι, cf. Arist. fr. 194; (ἀπέχεσθαι) ἀλεκτρυόνος λευνφᾷ,
Arist. fr. 195, Iam. VP 84; μὴ χρῆσθαι βαλανείῳ μηδὲ βαδίζειν τοὺς λεωφόρους, cf. lary
VP 83. Thus the intervening sentence (ἀπέχεσθαι) θνησειδίων also belongs to Aristotld
(cf. Hypomn., D.L. 33; missing in Boehm).
14 Tam. VP ὃς, p. 49.9ff: ἀγαθὸν of πόνοι, ai δὲ ἡδοναὶ ἐκ παντὸς τρόπου κακόν' a
κολάσει yap ἐλθόντας δεῖ κολασθῆναι 1 is nearly quoted by Arist. Protr. fr. 60 (lam. Promg
47. 25): τοῦτο γὰρ θείως οἱ ἀρχαιότεροι λέγουσι τὸ φάναι διδόναι τὴν ψυχὴν τιμωρίαν ahh
ζῆν ἡμᾶς ἐπὶ κολάσει μεγάλων τινῶν ἁμαρτημάτων, (Cf. Cic. Consolatio fr. 8 M., perhaps:
from Crantor: “‘Iuendorum scelerum causa nasci homines.’’) On Philolaus fr. 14, see € ‘
Ill 2 ἢ. 47.—Death as a peroixnots (lam. VP 85, p. §0.2) is Platonic (Ap. 4ος, Phd. 1196]
see Deubner on the Jamblichus passage); but it is tise understandable in the framework
the ancient doctrine of metempsychosis.
16 Por, VP 41 = Arist. fr. 196; also Plut. De Is. et Os. 32, Clem. Al. Strom. 4. 40,1".
16 Jam. VP 140: καὶ ἕν τοῦτο τῶν ἀκουσμάτων ἐστί: τίς εἶ, Πυθαγόρα; φασὶ vip
εἶναι ᾿Απόλλωνα ὝὙπερβόρειον (cf. ch. II 3, ἡ. 117).
17 VH 4.17; printed with Arist. fr. 196.
18 8.35: γῆρας καὶ πᾶν τὸ μειούμενον ὅμοιον" αὔξην καὶ νεότητα ταὐτόν. ὑγίειαν ~
τοῦ εἴδους διαμονήν, νόσον τὴν τούτου φθοράν" This sentence, like the one that precedes ἢ
(καὶ τῶν σχημάτων τὸ κάλλιστον σφαῖραν εἶναι τῶν στερεῶν, τῶν δὲ ἐπιπέδων κύκλον)
is excluded by Rose from fr. 195, but Delatte (Litt, 277, Vie 239) treats it as Aristotellan
The suspicious word εἶδος, in truce pre-Socratic fashion, means nothing more than the.
shape of the body.
19 An. post yabaa (DK s8C 1).
4. Acusmata
‘The traces of Aristotelian influence are still clearer in the τί μάλιστα
mip. ‘This is the same sort of wisdom as that attributed to the Seven
yes; for they too were trying to find not “what is good?’ but “what
most |pood]?’” (lam. VP 83). This is an excellent comment, and quite
irect from the point of view of the history of thought. Whatever the
w inay be about the authenticity of the sayings of the “Seven Sages”
κ΄ have collections of adages in the τί μάλιστα format),?° in the
atch for superlatives (τί φέρτατον, τί μάλιστον) there lies a very
wient kind of “wisdom,” to be found not only in the Contest of
lwner and Hesiod and in the Aesop legend, but also in Sappho (27 D.)
ul at the beginning of Pindar’s first Olympian. Now the designation
ithe Seven Sages in this passage as ἑπτὰ σοφισταί, an expression already
aoming obsolete in the fourth century, is attested precisely for
riatotle.24 What is more, in the passage we have cited twice above,
elian pives two τί μάλιστα sayings, one of which agrees with Iam-
clus ;# so that the similar ones in Diogenes Laertius must also be
‘tributed to Aristotle.?8
ila Clouv. sept. sap. 9.153c, D.L. 1.35, Stob. 1.1.29a, 1.4.7a, 1.8.40a, 1.18.1e; cf.
ἐἘ 1 deft and Ephorus ap. Diod. 9.26f. The saying σοφώτατον χρόνος, which occurs in
ja eeustext, is ascribed by Eudemus fr. 90 W. to Simonides. Lévy (Sources 4 n. 1) believes
8} the collection of sayings of Thales in D.L. 1.35 is presupposed at Iam. VP 83; but this
taprovable. The attempt of O. Brendel to reconstruct an original series of these sayings
WAL 41 [1010] 26f) is also problematic. Cf. B. Snell, Dichtung und Gesellschaft (1965) 103.
4 Fy 4, assigned to the On Philosophy because Aristotle dealt with the Seven Sages in
hat wink (tr. 3). soc. Antid. 235 attests that ἑπτὰ σοφισταί was an antiquated expression.
ἐ 4 Hyhorus ap. Diod. 9.26, Androtion ap. Aristides Or. 46, II 407 Dind.)—The attribu-
fan of this sentence to Aristotle yields a valuable chronological datum: πρότεροι yap
thins Hathiydpou ἐγένοντο. Hermippus (D.L. 1.42) names Pythagoras among the 17 who
Were, in various lists, included among the Seven Sages, but Dicaearchus (fr. 32 W.) does
get fee 1 litt.
111 4.17: πάντων σοφώτατον ὁ ἀριθμός, δεύτερος δὲ ὁ τοῖς πράγμασι τὰ ὀνόματα
Myews Ct lan. VP 82, p. 47.17: τί τὸ σοφώτατον; ἀριθμός: δεύτερον δὲ τὸ τοῖς
ἀμρήγμι τὴ ὀνόματα τιθέμενον. H. Steinthal, Geschichte der Sprachwissenschaft bei den
fitterhen tnd Romern | (Berlin, 18907) 153-168, tried to show that the second part was a
ἐμ 1 based on Pl Οὐαί, 416b-c. It is peculiar to have a “next best” given in the
ΗΠ 54 ol the τί μάλιοτα formulas. It may be that this second part was an addition. It is
ει cited by itself in the later tradition (Cic. Tuse. 1.62, Varro Ling. 8.7, Philo Qu. in
tien 1 40, 4.104, Decal. 23, lam. VP §6, Clem. Al. Eel. proph. 32.1, etc.; Boyancé, REG
a fives] so 72). Boyancé, REG 1941, 172, called attention to Herodotus’ statement
that it was Hlomer and Hesiod who ‘gave the gods their names” (2.53). Von Fritz
(HAT ium, 15) dates the acusma to the second half of the sth century, but admiration
af the γος deviser of names is already expressed in Aesch. Ag. 681-688.—Aelian
able the pronouncement on the mallow leaf (see ἢ. 40 below).
# Alaive, αν 68, on sphere and circle; and cf. Archytas DK 47A23a. Of course the
μη μεμα 1 distinction between στερεόν and ἐπίπεδον is not Pythagorean (above,
ἢ {40 uy); bur ifthe original formulation was τί κάλλιστον σχῆμα; κύκλος καὶ σφαῖρα,
the tiene precive terms would be adopted asa matter of course by later writers, including
ΑΗ Ηρ .
II, PYTHAGORAS IN THE EARLIEST TRADITION
Among this group we can even distinguish a pre-Aristotelian line
of tradition. Aristotle introduces a Pythagorean named Paron who
is otherwise completely unknown (DK 26): of μὲν σοφώτατον ἔλεγον
(τὸν χρόνον), ὁ δὲ ]]υθαγόρειος Πάρων ἀμαθέστατον os . (Phys. 222b17),
But in Eudemus’ more detailed account the story goes differently
(fr. 90 W.): ἐν ᾿ολυμπίᾳ Σιμωνίδου τὸν χρόνον ἐπαινοῦντος ὡς σοφώτατον;
a 3 Ὺ ~ ς 7 , ξ
εἴπερ ἐν αὐτῷ αἱ μαθήσεις γίνονται καὶ ai ἀναμνήσεις, παρόντα τινὰ τῶν
σοφῶν εἰπεῖν, τί δέ ὦ Σιμωνίδη, οὐκ ἐπιλανθανόμεθα μέντοι ἐν τῷ χρόνῳ ᾿
Eudemus cannot be dependent on Aristotle, since he sets the scene mort
elaborately—Olympia, Simonides, etc.—but what is a proper name t¢
Aristotle appears here as a participle: Πάρων-παρών. We cannot
however, follow Simplicius’ proposal to correct Πάρων in Aristoth
to παρών, for without the scene as Eudemus has it, the word would δι
meaningless. Both are following the same source, a written souret
which, with the accents unmarked, could be understood in ecitheg
of two ways, e.g. ITAPQN ΤΙΣ ITYOATOPEIOS EAETE,, 4
Thus we have a pre-Aristotelian proof of the high valuation placed on:
μνήμη by the Pythagoreans, and also of the Pythagoreans’ use of
proverbial wisdom in the contest of the ancient σοφισταί.
This shows that the entire section in Iamblichus is full of Aristotelian
material. It may be that the division of maxims into three categories ls
likewise Aristotle’s work. He was interested in logical distinctions,
and in particular in the early stages of conceptual definition.25 From
the lines of parallel transmission emerges quite an extensive body of
material.
What are the Isles of the Blest? Sun and moon. What is the Oracle
of Delphi? τετρακτύς- ὅπερ ἐστὶν ἡ ἁρμονία ἐν ἣ αἱ Σειρῆνες.25 Pythagoras
is the Hyperborean Apollo.?’ An earthquake is a mass mecting of the~
dead.?® The purpose of thunder is to threaten those in Tartarus, so that
they will be afraid.?® The rainbow is the reflected splendor of the sun,"
24 Diels, DK 1 217 n.: “‘Eudem folgte also einer anderen Lesart oder Ucberlieferung,"
This evades the problem. The source is likely to be a Sophistic-rhetorical writing like
Alcidamas’ Museum. Eudemus probably understood his source correctly, and the whole
chapter, DK 26, should be eliminated (Wehrli 106).
25 Cf. his understanding of the Pythagorean number symbolism as attempts at definition
(above, ch. I 2, n. 63). The equations he mentions like, “ marriage is §,” are at home among
the actistnata.
26 lam. VP 82; cf. below, ἢ. 154.
27 Above, ch. (13, nm. 107.
28 Ach. 4.17, The idea behind this may be that of a battle or struggle (Nilsson Τ 702 ἢν 1
referring to "Δ, 20.61fF).
Arist. An. post, gqlay.
Aci. 4.17; cf Anaximenes A7, AiR, and Xenophanes fr. 12.
4. Acusmata
‘Ihe sea is the tears of Cronus.3! The Great Bear and the Little Bear
are the hands of Rhea.3? The Pleiades are the lyre of the Muses, and the
planets are Persephone’s dogs. The ring of bronze when it is struck
ts the voice of a daemon entrapped in it;*4 and the ringing that people
wften hear in their ears is the voice of the κρείττονες. Old age is
ilecrease, and youth is increase; health is retention of form, disease its
dleatruction.?6 Friendship is harmonious equality.*”
‘The most just thing is to sacrifice, the wisest is number,?* the most
beautiful, harmony, the strongest, insight, the best (in the sense of the
tout desired), happiness, the truest, that men are wicked,®° the holiest,
a mallow leaf4? the most beautiful shapes, circle and sphere.*!
One ought to beget children, for it is our duty to leave behind, for
the pods, people to worship them.*® One should put on the right shoe
M Por. Ρ 41, Plut. De Is. et Os. 364a. Cf. the reference to the Adriatic as the “sea of
Crane,” Ap. Rhod. Argon. 4.327, and the localization of the myth of the castration of
Creme on the island of Corcyra (Timaeus FGrHist 566F79, Lycoph. 761, Orph. frag.
}47, 1.4) Cronus has a cave under Ocean, according to Por. De antr. nymph. 7 and Eust.
fa Dion. Per, 12.
# Avcording to the tradition the Little Bear was discovered by Thales (which means
that it was mentioned in the Ναυτικὴ ἀστρολογία which was attributed to him; Callim.
ἢ tue sat, with DLL. 1. 23). —In this context obviously belongs Procl. In Ἐπεὶ. 90.14:
wl Ad γε ΠΙυθαγόρειοι τὸν μὲν πόλον σφραγῖδα τῆς ‘Péas ἀποκαλεῖν ἠξίουν (referring to a
geal ἐμ Worn on the hand).
@ itelow, ch. [V 1.
δὲ Eliewe sentences are in Por. VP 41 (Arist. fr. 196), Plut. De Is. et Os. 32, Clem. Al.
Aten 4 s0.t; on the last, Cook, Zeus II 649ff; L. Parmentier, Mém. Acad. Belgique,
ΕἼ it lete., ser. IL, rz (1913) 31-61, with many parallel references. Cf. esp. Apollodorus
Hi atlit 224F ito, scholium and Eust. on I. 16.408 (ἤνοπι χαλκῷ); below, ch. V I.
# Acl 4.17; on κρείττονες see Hsch. s.v.
Ὁ 1}} 8.45; cf. above, n. 18. Similar is the correlation of the ages of man with the
Waine “Pythagoras” ap. D.L. 8.10, Diod. 10.9.5. But this is found as early as De
helehinadibus 4f, and in an allusive way in Heraclitus (fr. 100; cf. Reinhardt, Hermes
fuga, 2aHtt; Kirk, Heraclitus 204).
δ μον MGrHist 566F13 (cf. Pl. Leg. 7574, Hypomn. 33, lam. VP 162; Delatte,
Mt vor) In the same passage κοινὰ τὰ φίλων is given as a Pythagorean saying (cf. Pl.
i y? ain )
61 above, n. 22, Iam VP 82 has an additional item: τί σοφώτατον τῶν παρ᾽ ἡμῖν;
ἐβεμικ ἡ
hans ΡΒ); cf. above, ἢ. 14.
mA 417. Also, μαλάχης εἴργεσθαι, lam. VP 109; cf. Iam. Protr. 125.12ff; below, ἢ
tan The Neoplatonist Isidorus held to this precept (Dam. Isid. 125).
# Abwove, nm. 233 ch. IV 2.
tere the important thing is not the precept but the reason given for it (lam. VP 86).
Hate haw the same thought, philosophically expressed, Leg. 773¢: χρὴ τῆς ἀειγενοῦς
Mites dirdyerQar τῷ παῖδας παίδων καταλείποντα ἀεὶ τῷ θεῷ ὑπηρέτας ἀνθ᾽ αὑτοῦ
βηρμηδιδήναι, {{ 5 hardly likely that the simpler formulation found in the actsma was spun
nut ἢ the Platonic passage. Comparable is his idea that a human being is & τῶν κτημάτων
gute θεοῖν (Mid, Gab), Men are created as servants to the gods in Babylonian myth (ANET
ΒΕ)
II. PYTHAGORAS IN THE EARLIEST TRADITION
first,** not travel by the main roads (λεωφόροι) (no. 41), not dip one’s
hand into holy water (no. 44), not use the public baths (no. 45), and
not help a person to unload but only to load up.*# One should not
have children by a woman who wears gold jewelry (no. 21), not speak
in the dark (no. 51). One should pour libations over the handle of the
cup,® refrain from wearing rings with depictions of gods (no. 9), and
not “pursue” one’s own wife, since the husband, in receiving her as ἃ
suppliant at the altar, has taken her under his protection.*® One should
not sacrifice a white cock, because they are suppliants and sacred to the
god Men.*’ One should never give advice except with the best intent
(for advice is sacred) (no. 63),48 nor make a detour on the way to the
temple (no. 59). One should sacrifice and enter the temple barefoot
(no. 3), and in battle hold one’s place, so as to fall with wounds in the
breast.*® Eat only the flesh of animals that may be sacrificed; abstain
from beans; do not pick up food that falls from the table, for it belongs
to the Heroes.5° Abstain from fish that are sacred (including τρίγλη,
ἀκαλήφη, ἐρυθρῖνος, μελάνουρος); do not break bread;*! put salt on the
table as a symbol of righteousness. It is also forbidden to eat certair
. 43 A verse cited by a Polemon (of Ilium? Phot. Bibl. 553b20, no. VIII Nauck) adds that
in washing one’s feet one should do the left foot first. (Cf. Iam. Protr. 107.10, 114.9,
Menander fr. 97 Koerte.)
** This has the ring of a purely moral exhortation to πονεῖν. Méautis (78ff) refers to
stories about the underworld like Apul. Met. 6.18.3, in which one is forbidden to help .
a daemon one meets.
45 The thought is, of course, to avoid any intimate contact with the divine (Philostr,
VA 4.20).
46 Above, n. 12.
47 No. 13. The mention of the god Men certainly points toward Asia Minor as the
place of origin of this. Cumont (1942) would like to find here a direct contact between
Pythagoreans and the μάγοι, and points out that this prohibition is still observed today in
Tran (293f). But even if that be the place of origin, the white cock is connected with
Mithras and the sun, whereas the connection with Men (Shin) is Babylonian (Cumont
292, 285 n. 1). Therefore it must be a case of Anatolian syncretism rather than a direct
connection with Persia. (Later the cock is sacred to the sun: Iam. VP 147, Protr. 116,12.)
48 A widely known proverb. Cf. Hes. Op. 266, Epicharm. fr. 228 Kaibel, Ar. fr. 34,
Xen. An. 5.6.4, Pl. Ep. §.321c, Theag. 122b, Lucian Ind. 25, lam. VP 40.
* This well-worn maxim of military conduct may be interpreted not only in the
literary, physical sense, but in a mystical way. See Kirk, AJP 70 (1949) 384-393, come
menting on Heraclitus frr. 24 and 136, and M. Detienne, ‘Des confréries de guerricrs ἡ
la société pythagoricienne,” RHR 163 (1963) 127-131.
°° Aristotle (1D. L. 8.34) gives Aristophanes (fr. 305) as authority for this. On the topie,
Rohde, Psyche 1245 n. 1 = 202 n. 114 Eng. ed.
δ᾽ Basically synonymous with “Do not cat from a whole loaf” (above, n. 1; no. 30
Boehm), bread niust be cut with a knife, ina prescribed ritual manner.
This rationale excludes other, less pleasant interpretations, like the role of ἁλῶν
χόνδρος in mysteries of Aphrodite (Clem. Al. Profr, 2.14.2). Egyptian priests did not put
salt on the table (Plut. De Ts. ef Os, χ6 10.)
4. Acusmata
parts of the sacrificed animal, and there are many specific injunctions
about the time and the manner of sacrificing,®* and about rites connected
with death and burial. This is the context of the prohibition mentioned
hy Herodotus, of burying the dead in woolen garments.™
‘There are other rules, of just the same kind, in the tradition stemming
from Androcydes. For example, one is not to stir the fire with a knife
(μεν. 33), step over a yoke (no. 30), or sit on a bushel measure (no. 32).
(Mn rising one is to straighten the bedclothes and eliminate the traces
af one’s presence (no. 34),55 as well as destroying the marks of a pot
in the ashes (no. 35). One ought not to clean a chair with a torch
(1. 36)58 or to step, or make water, on nail parings (no. 48),57 and
should point a sharp knife in the other direction (no. 37), and not look
at oneself in a mirror with the help of artificial light (no. 52). On a
journcy do not turn around at the border.58
Ilermippus has some strange precepts: One should not pass by
where an ass is lying, and should avoid “thirst-causing water. 59
A cottin should not be made of cypress, for that is the material of Zeus’
wepter.°° The same sentence appears in a remarkable list of cult rules
in lamblichus (153-156); it combines quite trivial matters (“do not kill
a louse in the temple”) with commandments as important as the
prohibition of cremation (“in agreement with the Magi”). The
prohibition of “roasting what has been boiled” is attributed in the
prcudo-Aristotelian Problemata to “the mysteries. 6!
Whit we have here is a very mixed collection of sayings and maxims.
the threefold division, perhaps originated by Aristotle, is artificial and
ls not consistently followed. The rules and prohibitions regarding
ilaily life attracted most attention in ancient times; and there were two
upposite ways of interpreting them.
‘tam. VP ἃς; cf. 1§2.
ct. Inser. Délos 2180 = Sokolowski, .supp. 56; Lobeck 244f; J. Quasten, AJP
fey (1942) 207ff.—On Hdt. 2.81 see ch. II 3, π. 39.
"CL Por. ap. Stob. 1.49.59, Ar. Nub. 975f, W. Deonna, REG 42 (1929) 171, O.
Weinteich, ARW 28 (1930) 183-184.
™ The torch plays a role in cult, especially in ceremonies of purification; see Diphilus
to 120 Kock,
Not to spit on them,” Iam. Protr. 124.1; πρὸς ἥλιον τετραμμένον μὴ ὀμείχειν,
bot 8.47, kam. Protr. 115.19, Hes. Op. 727.
κ᾿ "μμινύες yap μετέρχονται, lam. Protr. 115.1, Hippol. Ref. 6.26.
© διψίων ὑδάτων ἀπέχεσθαι, Hermippus ap. Joseph. Ap. 1.164.
Hermippus ap. D.L. 8.10.
κι Problemata inedita ed. Bussemaker 3.43. Perhaps the reference is to the Dionysus
ritttal reflected in the Zagreus myth.
Material that belongs to the τί μάλιστα category in Acl. ΜῊ 4.17 (ἱερώτατον τὸ
ify μαλάχην φύλλον) appears asa prohibition in lamblichus (μαλάχην εἴργεσθαι, VP 109).
Il. PYTHAGORAS IN THE EARLIEST TRADITION
The account in Iamblichus, which goes back to Aristotle, leaves no
doubt that the precepts are intended to be taken literally (Iam. VP 86):
In some cases a statement is added as to why this is to be done®
(e.g., that one ought to have children in order to leave behind
another to worship the gods in one’s stead), but in other cases no
explanation is given. Some of the explanations added seem to be
ideally suitable, but others are far-fetched . . . The added, conjectural
explanations are not Pythagorean, but originate with persons who
introduced clever explanations from without,“ in an effort to give
a plausible rationale.
The most prominent topic of the acusmata is sacrificial ritual; and the
accounts of Herodotus, Isocrates, and Eudoxus speak of the special role
of ritual ἁγνεῖαι among Pythagoras and his followers.
Nevertheless, the prevailing view in antiquity was that what was
desired was not compliance to the letter but comprehension of the
deeper meaning. As early as Anaximander allegorical interpretation
was applied to this material as to Homer; and, when Aristotle used the
wotd ἐπισοφιζόμενοι, he surely had this method in mind. Our
principal source for the later interpretation is Androcydes. Here the
acusmata are regarded as αἰνίγματα, riddles, which clothe a lofty
wisdom in language unintelligible to the uninitiated.* As Clement puts
it, σφάλλεται μὲν 6 ἄπειρος καὶ ἀμαθής, καταλαμβάνει δὲ ὃ γνωστικός.
It is obvious that Androcydes must have exercised ἃ certain selectivity,
for not everything would fit together consistently. The injunction
καρδίαν μὴ ἐσθίειν means that one ought not to worry;*® but its come
88. ὀπιλέγεται «διὰ» τί δεῖ. The insertion of «διὰ» (a conjecture of Kiessling’s merely
recorded in the apparatus by Deubner) is essential. The omission would be an easy one in
a majuscule manuscript: ΕΠΙΛΔΕΓΕΤΑΙΔΙΑΤΙΔΕΙ͂.
84 ἐπισοφίζεσθαι is attested from the corpus Hippocraticum (LSJ); but εἰκοτολογία In
only found in later texts (ps.-Archytas p. 37.1 Thesleff).
65 Hdt. 2.81, Isoc. Bus. 28, Eudoxus fr. 36 Gisinger = 325 Lasserre == Por. VP 7.
86 Holk 21.
8? For the word aiviypara, see Tryphon, Demetrius, and Plutarch, n. 7 above, A
certain Hippomedon asserted (Iam. VP 87) that Pythagoras had given λόγους καὶ ἀποδείξειᾳ
for all his sayings, but that they had been lost through the carelessness of his successors,
Plutarch joins in the game of “solving” these riddles (Quaest. conv. 8.7, where the worda
of παλαιοί in section 2 obviously refer to Androcydes; ef DLL. 8.17). See also Philop,
De an. 116.29, Olympiod. Ia Plad, p. 8.22 Norvin, |
8 Clem. Al. Strom. 4.47.1 similarly, an anonymous writer (Phitarch?) ap. Stob,
3.1199: ὦστε, ., αὐτόθεν ἔχειν φῶς καὶ χαρακτῆρα τοῖς συνήθεσι τὸ φραζόμανοων,
τυφλὸν δὲ καὶ ἄσημον εἶναι τοῖς ἀπείροις ο΄,
No. τς Boehm: Demetrius, Plutarch, DWE. 821, ete eh Th 24.129... (ΑἥὕἸγἰκο fire 10g,
Gell. q.ccan, Ach Vi 4.171: ef bam. VP tog.
4. Acusmata
jutiion, the precept not to eat the μήτρα is not mentioned. Among the
fitbidden fish, the μελάνουρος (“blacktail’’) could be interpreted to
apply to men with “black” characters,’° but τρίγλη and ἀκαλήφη do not
Mippest anything interesting.
In any case, this new process of exposition and reinterpretation
wis quite inevitable, for in an enlightened period most of this material
would appear nothing but ridiculous. Thus the Pythagoreans were
led τὸ complain that Plato, Aristotle, and their pupils had appropriated
what was “fruitful” in the Pythagorean doctrine, with some little
revision, and left only the nonsense to them—all that was likely to
tnake the Pythagorean teaching seem absurd.?! Allegorical interpre-
tation, here as elsewhere, was the necessary means of adapting ancient
lave to new ways of thinking, and thus preserving its authority.
On the other hand, Aristotle’s report itself points the way to a
liflerent, more historical interpretation. He points out parallels with
popular belief, which we also can see reflected here and there in comedy,
and also with certain usages of the “barbarians’’ which had preserved,
av lie says, “even to the present day,” practices of their ancestors.”
‘the Alexander Polyhistor Memoirs, which, surprisingly, still ignore
the allegorical interpretation of the acusmata, add the significant
«umment that the abstinences they enjoin correspond to “the ones
prescribed by those who perform the initiatory rites in the temples.”
Along with the new type of interpretation comes an alteration in the
tenminology. In Aristotle, and only in his report, acusmata is the only
teria used for these precepts,”4 but outside the Aristotelian tradition
the prevailing word is symbola, though there can be no doubt that the
two words refer to the same thing.” Along with the word ἄκουσμα
tryphon, Plutarch (above, ἢ. 7), Arist. fr. 194. Androcydes may also have made
additioniv: the proverbial σιτίον εἰς ἀμίδα μὴ ἐμβάλλειν (Paroemiogr. gr. II 770) only makes
wae ita metaphorical interpretation.
Win [Ὁ $3.
At 1)... 8.34, Ar. fr. 305 is quoted (above, n. 50; Rose unjustifiably omits the citation
uf Atmophanes from fr. 105).--τ-.καθάπερ ἔτι καὶ νῦν of βάρβαροι, D.L. 8.35 (cf. Iam.
ji NG).
Hypomn. 33 (cf. lam. VP 138). This excludes the “Androcydes” interpretation of
tle taloo on beans and that on blacktail (both mentioned in Hypomm. 33); and this also
proluably indicates that “Androcydes” should be dated later than the Hypomnemata
(uel century wc.). The “Androcydes” interpretation of the taboo on beans—that one
wipgld not to tiuke part in politics—is hardly pre-Hellenistic. (A different interpretation,
haailoon the same reason, that the bean was used to vote with, is found in Arist. fr.
vy 191. 5.14: ὅτι ὀλιγαρχικόν.)
hain, VP Ratt, τ40. Elsewhere the word ἄκουσμα has the sense “ musical entertainment”
(Men Aten δια. γι. Arist. Pol. 1336b2, lam. UP 245, etc.).
ΒἍΤΟΙΚ (ΠῚ !
II. PYTHAGORAS IN THE EARLIEST TRADITION
goes the designation of the members of the Pythagorean society as
ὁμάκοοι and ὁμακοεῖον for their meeting house.”* On the other hand,
the word σύμβολον carries with it the suggestion of a “symbolic”
interpretation. The word is not a late addition, however,’’ but carries
another implication as well, prior to any “symbolic” exegesis. In the
realm of mystery religion, σύμβολα are “passwords” —specified for-
mulas, sayings, ἐπῳδαί, which are given the initiate and which provide
him assurance that by his fellows, and especially by the gods, his
new, special status will be recognized.’® The acusmata or symbola still
have this function as late as Lucian’s parody: τούτων yap ἂν μεμνημένον
ἐλπίδας ἔχειν τῆς εἰς τὴν νῆσον ἀφίξεως, says Rhadamanthys, in the True
History, to Lucian as he leaves the Isle of the Blest.7® Aristoxenus
mentions σύμβολα as “ passwords ’”’ used by the Pythagoreans.®° The pen-
tagram, too, was a symbolon in this sense; it was thought from the most
ancient times to have a secret power and significance.8! But when once
the allegorical and symbolic interpretation of the Pythagorean sayings
had gained a foothold, it was inevitable that the word σύμβολον be
understood in this sense, whereas ἄκουσμα had no such obvious
meaning and fell out of use.
The riddle as a literary form (γρῖφος) is very old, and is used in
the promulgation of oracles. There can be no doubt, though, that the
acusmata are, rather than simple, commonsense wisdom in abstruse
8 ὁμακοεῖον Nicom. (lam. VP 30 = Por. VP 20), lam. VP 74, 185. Altered to ὁμακόιον,
Olympiod. In Phd. p. 9.10 Norvin, and copied by Schol. Pl. Phd. 61c, Olympiod. In
Alc, p. 132.12, Eust. p. 856.63f; ὁμάιον, Hierocles In CA 27 (p. 484 Mullach), Eust. loc.
cit.—opaxoot, Iam. VP 73 (Timaeus? above, ch. II 1, n. 37). The compound, with ὅμο-
and the Doric form, indicates that an original term has been preserved. On συνήκοος
(Pl. Leg. 711e), see ch. I 4.
7 Pace Holk 15ff.
78 'W. Miiri, “Symbolon: Wort- und sachgeschichtliche Studie,” Beilage to Jahresber.
d. St. Gymn. (Bern, 1931) 39ff; Boyancé, Muses «τῇ; cf. the Orphic papyrus DK 1B23,
the Eleusinian σύνθημα, Clem. Al. Protr. 2.21 (σύμβολον -- σύνθημα, “ password;” Eur.
Rhes. §72f). Cf. Plaut. Mil. 1016: ‘“‘cedo signum, si harunc Baccharum es.”
9 Ver. hist. 2.28, Nock, AJA 1946, 153.
8° Fr. 43 W.: (Pythagoras) ef τινα πύθοιτο τῶν συμβόλων αὐτοῦ κεκοινωνηκότα, εὐθύς
τε προσηταιρίζετο καὶ φίλον κατεσκεύαζεν.
*! Lucian Laps. 5; Schol. Lucian p. 234.21 Rabe; Schol. Ar. Nub. 609. The pentagram
has been very widely used as a magical symbol, from the times of the carly oriental
civilizations. See R. Eisler, Weltenmantel und Himmelszelt 1 (Munich, roto) 304f; j.
Leite de Vasconsellos, Signum Salomonis: Lstudo de etnografia comparativa (Lissabon,
1918); Cook, Zeus ΠῚ 341 1. 3; ΝΜ. Deonna, Bull. de PAss. Pro Aventico 16 (1954) 44;
C. Lascaris Comneno and A.M. de Guadan, Rev. di filos. ας (1956) 181-207 (with the
comment of W. Burkert, Philologus 1961, 230 n. 4); Stapleton, Osiris 1958, 126 39;
De Vogel 292-299.
4. Acusmata
form,® ancient magical-ritual commandments.® It is not possible here
to offer a complete analysis, but we may try to establish a few points of
reference.
There are obvious coincidences with rituals of Greek mystery
cults, and they even share whole series of commandments. For
example,
(a) The initiate at Eleusis must fast, avoid baths, abstain from
domestic fowl (and the cock is especially named), as well as ἰχθύων καὶ
κυάμων ῥοιᾶς τε καὶ μήλου, καὶ ἐπ᾽ ἴσης μεμίανται τό τε λεχοῦς ἅψασθαι
καὶ τὸ θνησειδίων.83
(Ὁ) At the Haloa, one was forbidden to eat ῥοιά, μῆλον, ὄρνιθες
κατοικίδιοι, wd, as well as the fishes τρίγλη, ἐρυθῖνος, peAdvoupos,
κιίραβος, γαλεός. 88
(c) Preparation for the katabasis in the Trophonius cult included
avoidance of warm baths, τρίγλη, τρυγών, μελάνουρος, and one must wear
a linen garment.*¢
(d) The μάγοι, καθαρταί, ἀγύρται, ἀλαζόνες against whom the Hip-
pocratic work On the Sacred Disease speaks, forbid their patients baths,
τρίγλη, μελάνουρος, κεστρεύς, ἔγχελυς, certain kinds of meat, various birds
including the cock, and spices. They may not wear any black garment,
lic on a goatskin, lay one foot or one hand over the other; and there
are other such purificatory measures.°?
(c) In Delos, the worshipers of Zeus Cynthius approach him barefoot
and clad in white. They abstain from sexual intercourse and from meat,
and do not wear iron rings, keys, belts, purses, or weapons.*®
"2 Maddalena thinks that the acusmatici misunderstood Pythagoras’ metaphorical
linguage (362). Von Fritz (SBMii 1960, 16f) regards the allegorical sense as primary in
ζυγὸν μὴ ὑπερβαίνειν (but what of the variant odpov in Plut. Quaest. conv. 727¢, Quaest.
Rom. 290e, Hippol. Ref. 6.27.4?), and also in καρδίαν μὴ ἐσθίειν. (On this, see below, n.
110; above, n. 70.)
Ks“ Ritualgesetze,”” Rohde, Q 109; cf. Lobeck go2f, 248f; Nilsson I 703-708. “Genuine
taboos,” Burnet EGP 96. Boehm gathers comparative material. Delatte, Litt. 287f,
tuvors the ethnological method, though elsewhere he hesitates (Vie 186f: “‘trop uniforme
pour étre toujours vrai’’).
"4 Por. Abst. 4.16, Hymn. Hom. Cer. 50. The avoidance of λεχώ and θνησείδια is also
prescribed at Hypomn. 33; cf. also n. 102.
86 Schol. Lucian p. 280.23 Rabe.
“6 Ccratinus fr. 221 Kock (Ath. 7.325¢), Paus. 9.39.5, 8. The proscription of meat-
eating and baths is also in the Paris magical papyrus PGM no. IV 736.
"7 Hippoc. VI 354f L., cited by Delatte, Vie 232; Boyancé, Muses 107. A connection
of these καθαρταί with Pythagoreanism is not to be excluded; M. Wellmann, Die Frag-
mente der Sikelischen Arzte (Berlin, 1901) 29.1, conjectures that Empedocles and his school
are the persons meant.
“4 Taser. Deélos 2529 9 Zichen LS gt,
Il. PYTHAGORAS IN THE EARLIEST TRADITION
(f) The sacred law of Lycosura commands that one sacrifice barefoot,
wear white clothing, and neither rings nor other gold ornaments.®*
(z) In the Temple of Asclepius one may not pick up anything that
falls to the floor (lam. VP 126), and one who undergoes the sleep cure at
Pergamum may not wear ring, belt, or gold ornament.*°
It may be that in a few cases Pythagorean ritual was adopted in
late times by various cults,®! but in general the latter are independent
of Pythagoreanism and older than it. Some taboos are attested from
an earlier date,®? and a good many are widely spread folk tradition.”
Above all, the form of such authoritatively prescribed commandments
and prohibitions, which are not supposed to be understood but merely
obeyed, is primeval; it is entwined in the very roots of religious ritual.
In what one does and does not do is manifested the identity of the
group, the membership of the members and the exclusion of outsiders.
The more selective the society, the more careful are the “taboos.”
Fasting, abstention from particular foods, and rules of sexual behavior™
play an important role. It is of first importance that the “wise man’”’
—the priest, the hierophant, the shaman—who claims a special position
in the social organization, gain and maintain, through a special ascetic
regimen, the special powers that belong to him. To this extent, the
acusmata also have their connection with the Pythagoras legend and
its cultic bases. The saying that Pythagoras was the Hyperborean Apollo
is expressly attested as an acusma (Iam. VP 140).
Pythagorean silence and secrecy should also be seen in the context
of cult and ritual. To be sure, the secrecy of Pythagorean doctrine
89 SIG’ 999 (Delatte, Vie 231f); similarly in Andania (SIG® 736): no shoes, no gold,
linen clothing; in Ialysus (SIG? 338): no shoes as in the mysteries of Demeter (Callim.
Cer. 124.)
9 Sokolowski, 14. Cf. also L. Deubner, De incubatione (Leipzig, 1900) 148}
®t The prohibition of beans is found in cult regulations of the imperial period: Rhodes
(Sokolowski, supp. 108), along with the prohibition of ἀφροδίσια and καρδία;
Smyrna (Sokolowski, 84) along with the prohibition of eggs. Perhaps the Titans
were somehow brought into this context, but the text cannot be restored with confidence.
Probably this is a case of neo-Pythagorean influence (Nock, HSCP 63 [1958] 415-421);
an ancient ritual can of course be understood in late times as Pythagorean.
92 Hes. Op. 727 (above, n. §7); Hymn. Hom. Cer. 50 (above, n. 84); cf. also the Ζελλοὶ
ἀνιπτόποδες χαμαιεῦναι at Dodona, Il. 16.234.
98 Cf. above, n. 43.
94 Complete celibacy is ascribed to Pythagoras at 1.L. 8.19; but this is contradicted by
the traditions that include his wife and children (above, ch. II 2). The acusuta cited in n.
42 (above) impiies the desirability of marriage, as do the sayings of Theano in D.L. 8.43
(cf. Hdt. 1.8; A. Raubitschek, REM too [tos7] 1.20}, and Tam. VP 132, Stob. 4.23.53.
Pythagoras demands strict monogamy according to Philoser. 1A 1.13, Tam. VP 48, $0,
A.P. 6.43. Special regulations as to dict and regimen, in relation to sexual intercourse,
are given by Aristox, fro ag W., DLE. 8.9, and Diod. -ro.g ΜΠ Clinias,” a character in
Plat. Quaest, conv, ὁ 440, is negative on the matter,
4. Acusmata
later was misused by forgers as license to “discover” more and more
Pythagorean writings.®> But the testimony of Aristotle and Aristoxenus,
which proves the existence of Pythagorean ἀπόρρητα, cannot be ig-
nored.% Aristotle's testimony has to do with the intermediate position
of Pythagoras between man and god. All mysteries have secrets; the
ritual is interpreted in a ἱερὸς λόγος which may not be disclosed to the
uninitiate,*? and the initiate also learns secret passwords, σύμβολα,
συνθήματα. All kinds of societies that are bound together by cult have
their esoteric aspect—even political clubs, trade guilds, and those of
physicians.°8 Among the Pythagoreans the practice of remaining
silent for long periods—one of the most effective means of attaining
inner composure—is to some extent a continuation of the practice of
shamans and yogis.°® The ὁμάκοος is not supposed to speak but to
hear, and as early as Isocrates and the comic poet Alexis Pythagorean
silence was proverbial? A five-year period of silence as a test before
acceptance in membership is attested by Timacus.!" The practice was,
also, to avoid using Pythagoras’ name
Aristoxenus mentions that Epaminondas called the Pythagorean
Lysis, who was his teacher, “ father.’’1°* Ludwig Edelstein brought this
into connection with the precept of the Hippocratic Oath that the
physician must regard his teacher as his father and the teacher’s sons
as brothers.!° But the same usage is especially prevalent in the
nrysteries ;!°5 whoever has led the candidate to be initiated becomes his
father. The Pythagoreans, too, form a “brotherhood,” in accordance
% Burkert, Philologus 1961; below, ch. III 1.
96 Arist. fr. 192 = Iam. VP 31, Aristox. fr. 43 W. = D.L. 8.15: μὴ εἶναι πρὸς πάντας
πάντα βητά. Further references in Zeller I 409 n. 2. On the alleged secrecy of Pythagorean
mathematics, see below, ch. VI 3. The whole tradition of secrecy is rejected as a late
invention by (among others) Maddalena 80f, and G. Boas, “Ancient Testimony to Secret
Doctrines,” Philos. Rev. 62 (1953) 79-92. The latter completely overlooks the attestation
of Aristotle and Aristoxenus. The ritual background of Pythagorean secrecy was brought
out by Zeller. loc. cit., Rey 112f, and Mondolfo in ΖΜ 414 ἢ. 2.
"7 ἱερός is used here in the sense of ody ὅσιον λέγειν, cf. Hdt. 2.61f.
8 Cf. Hippoc. Jusj. and Lex, Ar. Nub. 140ff; Gigon, Ursprung 130f.
"Cf. also O. Casel, De philosophorum graecorum silentio mystico (Giessen, 1919), G.
Mensching, Das heilige Schweigen (Giessen, 1926).
09 Tsoc. Bus. 29, Alexis fr. 197 Kock, and Dicaearchus ap. Por. VP το.
0! FGrHist §66F13 = D.L. 8.10, lam. VP 72, etc. The technical terms ἐχεμυθία and
ἐχεμυθεῖν are found in Ath. 7.308d, Plut. Numa 8, Quaest. conv. 728e, Lucian Gall. 2,
lam. VP 94, ἐχερημοσύνη at Iam. VP 246.
Tam. VP $3, 88, 25§. The hicrophant at Eleusis is not called by name (P. Foucart,
Les mysteres d’Eleusis [Paris, 1914] 173ff).
103 Aristox. fr. 18 --- Iam. VP 250, Diod. 10.11.2, Nep. Epamt. 2.2, Plut. De gen. §83¢.
4 OMG: 11, p. 4.9ff; L. Edelstein, The Hippocratic Oath (Baltimore, 1943) 34fF.
106 A. Dieterich, Hine Mithrasliturgie, 19239, 146-149; 1 Cor. 4.15. For the adoption
of Heracles and the Dioscurt in [leusis, see Plat. ‘Thes. 33. On “Chaldeans,” Diod.
2.20.4.
I. PYTHAGORAS IN THE EARLIEST TRADITION
with the ancient custom of colleagues bound together in a cult. We
are even told that Parmenides adopted Zeno.1%
That the “Pythagorean life’’ developed from living custom, with
all its complexity and paradox, rather than from clearly articulated
doctrine, can be seen very clearly in the rules about abstinence from
meat.1® The self-evident corollary of the doctrine of metempsychosis
would have to be complete vegetarianism. Empedocles drew this
conclusion, and according to Eudoxus, Pythagoras was not only a
vegetarian but avoided any association with butchers and hunters.!°8
Aristoxenus, however, asserted that he only avoided eating plow oxen
and rams, but was especially fond of the meat of tender young kids,
sucking pigs, and cockerels.°® Hints of Athenian ways here are
intended to lessen the absurdity of Pythagoreanism, for the benefit of
the reading public. There are other remarkably persistent traditions
that seem to know nothing of Pythagoras as vegetarian. The famous
sacrifice of an ox to celebrate a geometrical discovery is attested by a
(probably) fourth-century source," and the tradition is hardly later
that it was precisely Pythagoras who introduced the meat diet for
106 Apollodorus FGrHist 244F30; πατήρ, Pl. Soph. 241d, De Vogel 240f.
107 Cf. Zeller I 403f, and esp. Haussleiter 97-157.
108 Fr, 36 Gisinger = 325 Lasserre = Por. VP 7. (The “impurity” of hunters and
butchers does not exclude the eating of meat, but is actually evidence for it; see Meuli,
Opferbr. 228; Agatharchides GGM I 154, on the Troglodytes.) Vegetarianism is charac
teristic of the Πυθαγορισταί and of Diodorus of Aspendus (below, ch. II 5). See also
Onesicritus FGrHist 134F17 (Strabo 15, p. 716), Callim. fr. 191.61f Pfeiffer. For the
story that at Delos Pythagoras only worshiped at the altar of Apollo Genetor, where
bloody offerings were not made, see, e.g., Iam. VP 25, 35, Cic. Nat. 4. 3.88, Clem. Al.
Strom. 7.32, Macrob. Sat. 3.6. Arist. fr. 489 == D.L. 8.13 and Timaeus FGrHist 566F147
== Cens. 2.3 mentioned the altar, but it is not certain that they brought Pythagoras into
connection with it (Delatte, Vie 177). Among later writers, see Ov. Met. 15.75ff (with
reminiscence of Empedocles) and Sotion ap. Sen. Ep. 108.17.
109 Prr, 2§, 28, 29, D.L. 8.20, Diogenes Antonius ap. Por. VP 36, Iam. VP 150 (where
cockerels are explicitly mentioned). Boyancé pointed out the connection of the sacrifice
of rams and pigs with Attic ritual, REG 1939, 4off. On the plow ox cf. Wehrli on Aristox.
fr. 29a, Haussleiter 116.3-4, Meuli, Opferbr. 275ff. With Aristox. fr. 27 are connected
D.L. 8.19 and Iam. VP 98 (Boyancé, loc. cit.); the former mentions woolen clothing.
At Eleusis it was said to be the command of Triptolemus γονεῖς τιμᾶν, θεοὺς καρποῖς
ἀγάλλειν, ζῷα μὴ σίνεσθαι (Por. Abst. 4.22 == Kenocrates fr. 98), and Aristoxenus ascribes
similar regulations to the Pythagoreans (D.L. 8.23, Por. VP 39, Iam. VP gof).
10 The basic testimony is an epigram by a certain Apollodorus, who should perhaps
be identified with the philosopher of Cyzicus (DK 74) and dated earlier than Epicurus
(below, ch. TIF 1, n. 41). The means chosen to avoid this testimony was to assert that
Pythagoras had sacrificed an ox of dough (Diogenes Antonius ap. Por. VP 36; Greg.
Naz. Ep. 198, Migne 37.324, where σταίτινος is misread as πήλινον), This is obviously
transferred to Pythagoras from Empedocles (Ath, 1.3¢, Favorinus ap. DL. 8.53, Suda s.v.
Athenatos; cf. Philostr. 1A 1.0.2; Haussleiter 162, Delatte, Vie 174). For similar substicute
offerings, sce Edt. 2.47.
4. Acusmata
athletes.4! Nicomachus smoothed out the contradictions of the evidence
by use of the reports about degrees of membership within the society:
the θεωρητικοί practiced complete abstention, whereas the πολιτικοὶ
ἢ ἀκουσματικοί ate meat, though “seldom.”"* But the very Pythago-
rcans whom we must suppose to be “‘acusmatics,”’ the “ Pythagorists,”
Diodorus of Aspendus, and the Cynic-influenced Onesicritus, em-
phasize the radical formulation ἐμψύχων ἀπέχεσθαι, while Aristoxenus
seems to be thinking rather of “theoretical” Pythagoreans, who still
like meat.
To judge by Aristotle's testimony, the acusmata did not contain any
simple prohibition of the eating of meat, but various specific precepts:
the heart and womb of the animal, and perhaps also other similar
parts,44 may not be eaten. Here, as also in the prohibition of killing
ἃ white cock or of eating the flesh of animals that have died a natural
death," it is taken for granted that other kinds of meat will be eaten.
Special treatment of certain parts of the slaughtered animal was part of
sacrificial ritual from very early times. Karl Meuli has shown that
the horror of death and the reverence for life manifest themselves in
these usages and also express themselves in an attempt at recompense,
that is in the rebirth of the slain creature." The heart has a special role
in Greek ritual."6 This is reflected in the Orphic myth in which Athena
1 Por, Abst. 1.26 = Heraclides fr. 40 W. (though it is impossible to determine how
much of the context goes back to Heraclides), Favorinus ap. D.L. 8.12, Por. VP 15. In
order to explain away this evidence, some postulated a different Pythagoras: D.L. 8.13,
%.46 (here he is a Phliasian!), Plin. HN 23.121, Iam. VP Ὡς (“son of Eratocles”): τούτου
δὴ καὶ Ta ἀλειπτικὰ συγγράματα φέρεται, οὐ καλῶς εἰς Πυθαγόραν τὸν νημάρχου τούτων
ἀναφερομένων. There was in circulation, then, a coach’s handbook under the name of
l’ythagoras. The tradition must have been formed before Pythagorean vegetarianism
was firmly established. Scholars have referred, in discussing this matter, to the famous
athlete Milo of Croton, who was regarded as a Pythagorean (Haussleiter 124f; as Pytha-
yorean, Iam. VP 104, Aristox, fr. 18 = Iam. VP 249, Strabo 6.263; as a heavy eater,
Ath. 10.412ef). Iccus of Tarentum was a famous trainer (a Pythagorean according to
lam. VP p. 144.6; DK 25; Wuilleumier 566). The boundary between the “Pythagorean
lite” and rational diet fluctuates (cf. ch. ΠῚ 3).
Tam. VP 107f, 150. (The allusion in 108 to legends about Pythagoras persuading
wild animals not to eat meat points to Nicomachus as the source of the passage; above,
ch. IH 3, Rohde, Q 143f.) The differentiation according to classes of membership is
accepted by Haussleiter 119, Vlastos, Philos Q. 1952, 110 ἢ. 62, von Fritz, SBM 1960,
14. Eubulus, Περὶ τοῦ Μίθρα ἱστορία (ap. Por. Abst. 4.16) distinguishes three classes among
the Magi; cf. below, ch. IIs.
"lt Arist. fr. 194 (above, n. 69). Add ἐγκέφαλον μὴ ἐσθίειν Iam. VP τοῦ, Iam. Protr.
123.14, Plut. Quaest. conv. 635¢. There is a different list in Por. VP 43.
"4 Θνῃσείδια: Acl. 4.17, Hypomn, 33; cf. above, n. 13.
ΠΣ Meuli, Opferbr., esp. pp. 185 fF.
116 It was cut out of the living animal and laid on the altar still beating (Galen Place.
Hipp. et Plat. V 238 Και; Etym., magn, καρδιωσάμενοι, Suda, ιν. xapSiwodpevos).
Π. PYTHAGORAS IN THE EARLIEST TRADITION
saves the heart of Dionysus Zagreus when he is torn to pieces by the
Titans and hides it in a sacred chest," and also in the speculations of the
physical philosophers, according to which the heart is the first of the
organs to grow."38
Animal sacrifice was the focal point of the traditional ἘΠῚ
that is of the official cult of the polis, and to renounce it would have
been more than religious reform. It would have meant a complete
overturn of traditional ways. It is interesting that this is the goal of
Zarathustra's gospel." As far as we can judge, the Pythagoreans sought
to compromise the matter; an acusma asks, “What is most just?”
and answers, [Ὁ sacrifice.” An accommodation of the doctrine of
metempsychosis and the traditional way was found, because it had to
be found. “The only animals into which the souls of men do not enter
are those which may, according to sacred law, be sacrificed. Therefore,
those who are allowed to eat meat may eat only of those animals that
may be sacrificed, but of no others.’’!2°
In the late compromise, it seems that the ancient cult practice is
still influential. Originally, and for a long time, abstinence was only
a preparation for the sacred meal, so that omophagia and vegetarianism,
different as they seem, are complementary.1#! In the mysteries of Demeter
and Dionysus the most important sacrificial animals are sucking pigs,
cocks, and kids,!?? the very animals of whose meat, according to
Aristoxenus, Pythagoras was especially fond. Perhaps, then, this
statement was not manufactured out of whole cloth, but was a ratio-
nalization of ritual. One report says of the Pythagoreans, “Throughout
their lives they abstained from eating meat; and when, in their own
stead, they made the prime-offering of an animal to the gods, they
117 M. Tierney, “A Pythagorean Tabu,” Mél. E. Boisacq (Brussels, 1935) 317-321,
referring to Clem. Al. Strom. 2.17.2, 2.22.4 and Firm. Mat. Err. prof. rel. 6.1ff. That
Pythagoreanism is here directly dependent on Orphism is possible but not definitely
provable.
118 Empedocles A 84, Plut. Quaest. conv. 6356; at 636d the Orphic prohibition of eggs is
interpreted in the same way; it is attested for the Pythagoreans only at Hypomn. 33.
119 The fight against the sacrifice of the cow, Yasna 44.20; 92; cf. above, ch. IT 2, n. τό.
120Jam. VP ὃς, Por. Abst. 1.26; cf. also Boyancé, REG 1939, 52 n. 2. Though this
interpretation may be a counsel of desperation, it need not on that account be secondary,
for the awkward situation was there from the beginning. There has, however, been an
interpolation in the excerpt from Aristotle; καθήκει is not used impersonally by Aristotle.
— Another way out was to consider sacrifice as “justified execution” (Por. ap. Stob. 1.49.9;
cf. Pl. Leg. 8702).
121 (Cf, Eur. fr. 472.
122-‘The cock and the mysteries of Demeter: Por, Abst. 4.16; sucking pigs sacrificed at
Heusis: Ar. Pan 774, with scholia; kids: the ἔριφος ety γάλα of the Gold Plates; cf also
GRAS 7 (19066) yy.
4. Acusmata
would just taste it, so that in truth they lived untouched by such
things, ’'228
The famous, or notorious, taboo on beans is found in similar
contexts. It is attested by Aristotle, Heraclides, Callimachus, and
indirectly by Empedocles;!*4 only Aristoxenus denies 11.125 This was a
favorite target of those who would mock Pythagoreanism. As early
as Aristotle several alternative explanations of the puzzling aspects of
it were available. The interpretations that are most closely related to
yenuine Pythagoreanism are those which connect beans with the
doctrine of metempsychosis. Along with Aristotle’s report that beans
ure like the “gates of Hades,” since they alone among plants do not
lave joints,126 we have a remarkable couplet in the scholia to Homer :137
ψυχῆς αἰζηῶν βάσιν ἔμμεναι ἠδ᾽ ἀναβαθμὸν
ἐξ ᾿Αίδαο δόμων, ὅταν αὐγὰς εἰσανίωσιν.
It is through bean blossoms that souls return to earth for their rein-
carnation. Varro says that the souls of the dead are in the beans;128
and there are several variants of the conception that a soul emerges from
1 plant, or a bloom, to enter into a human body.”® This makes com-
prehensible the legend that Pythagoreans refused to walk through a
4 Por, Abst. 2.28, which is not from Theophrastus according to J. Bernays, Theo-
phrastos’ Schrift tiber Froémmigkeit (Berlin, 1866) 119, and W. Potscher, Theophrastos [epi
εὐσεβείας (Leiden, 1964) 176. Cf. Arist. fr. 194 (D.L. 8.19): "AptororéAns δέ φησι καὶ
μήτρας καὶ τρίγλης ἐνίοτε. Here the word ἐνίοτε means “at certain times.” See also Iam.
ΕΓ 85: the acusmata are concerned, above all, περί τε θυσίας καθ᾽ ἑκάστους τοὺς καιροὺς
πῶς χρὴ ποιεῖσθαι. Cf. Burnet, EGP ος.
4 Arist. fr. 195 = D.L. 8.34, Heraclides fr. 41 (surely to be attributed to the book
Hepi τῶν Πυθαγορείων, although Pythagoras is not named), Callim. fr. 553 Pfeiffer;
ilso Emp. fr. 141, and the frequently cited verse fady τοι κυάμους τε φαγεῖν κεφαλάς τε
τοκήων (references in Orph. frag. 291 Kern, Delatte, Faba 36 n. 2). The subject is fully
canvassed by Boehm 14ff; R. Wiinsch, Das Frithlingsfest der Insel Malta (Leipzig, 1902)
31-46; Haussleiter 407ff; Delatte, Faba; M. Marcovich, Philologus 108 (1964) 29-39. It
is well known that the bean in question is a kind of European vetch (Vicia faba); the beans
used as vegetables nowadays are of American origin.
125 Fr, 25 W.; cf. above, ch. II 1, n. $7.
"6 Arist. fr. 195 = D.L. 8.34. On the word dydvarov see Por. De antr. nymph. 19
(Iclatte, Litt. 36ff, Faba 36ff; R. D. Hicks is wrong in suggesting a lacuna before ἀγόνατον,
in his Loeb edition of D.L.).
27 Schol. T If. 13.589. no. VI Nauck.
28 Ap. Plin. HN 18.118. At the Roman festival of the Lemuria, one throws beans
behind him, at night, “et dicunt se Lemurios domo extra ianuam cicere” (Varro ap. Non.
p. 135.15 M.; somewhat differently Ov. Fast. 5.43 5ff).
129 Attested as carly as Aeschylus, in the story of Glaucus, fr. 28f; see Paus. 9.22.7,
Philostr. VA τος, Lévy, Leg. 178ff, Ov. Fast. §.231ff, H. Usener, ΚΙ. Schr. 1V (Leipzig,
1913) 128ff.
I. PYTHAGORAS IN THE EARLIEST TRADITION
bean field in bloom;?*° but it scarcely brings us closer to understanding
the real reason for the importance of beans.
We dare not take too lightly the rationalizing, physiological
explanation, based on the difficulty of digesting beans.!*' The seer or
sage is very sensitive to small physical disturbances. It may have been
seen by the ancients, even before the discovery of the chemistry of
proteins, that of all vegetable diets one of beans is most like one of
meat. A contributing factor in this recognition may have been that a
certain amino acid, present in beans, can provoke strong allergic
reactions in some persons.!% In any case the peculiarities of beans had
obviously, from early times, been exploited in cult and myth. The most
interesting of the explanations of the bean taboo claim amazing
similarities between bean and man: the blossom of the plant, or the
bean itself, are transformed through certain procedures into human
form, or into the form of parts of the human body," they remind of
the genitalia! and smell like semen. At the origin of the world, bean
and man emerged from the same primeval slime.48* And when we
look for the origins of these prolific fancies, we are once more led to
the mysteries. Demeter, says Pausanias,!8* gave mankind all the pro-
ducts of the earth, with the exception of beans; and “whoever has
witnessed an initiation at Eleusis or read the so-called Orphic writings,
knows what I mean.” We are denied this knowledge. It may be that
beans were eaten in a ritual meal, and if so, Aristoxenus could be
180 Myllias and Timycha; Neanthes FGrHist 84F31 = Iam. VP τοι.
131 The φυσῶδες of beans is often emphasized, as in the joke used by Heniochus, fr. 4
(Ath. 9.408a), ἔτνος κυάμινον διότι τὴν μὲν γαστέρα φυσᾷ τὸ δὲ πῦρ ov (γῇ). Cf. Hippoc.
Vict. 2.45; the collection of references in Delatte, Faba 54ff, and Boyancé, Muses 111.2.
Delatte derives the taboo on beans entirely from their effects on digestion, but with too
easy a transition from φῦσαι to animism. The mythical account cannot be deduced from
the physiological facts —Amphiaraus avoided beans for the sake of his art of divination
(Geop. 2.358); cf. Hdt. 4.184, Cic. Div. 1.62.
132 Capparelli I 187 ἢ. 3, II 825ff, with references.
138 Heraclides fr. 41 (the bean in the coffin), Diogenes Antonius (Por. VP 44 = Lydus
Mens. 4.42), Hippol. Ref. 1.2.15 (πε buried bean blossom), also in a Demotic papyrus
(T. Hopfner, Offenbarungszauber 1 [Leipzig, 1921] 135).—Transformation into human
blood: Lucian V. auct. 6. See also Marcovich, cited above, ἢ. 124.
134 Arist. f-. 195, Gell. 4.11.10.
1385 Por. VP 44, Hippol. Ref. 1.2.14.
136 7.37, 45 cf. 8.15.3f, Por. Abst. 4.16. There is some sort of connection between this
and the ἥρως Κυαμίτης at Athens. Hymn. Orph. 26 prescribes, as an offering for Ge,
θυμίαμα πᾶν σπέρμα πλὴν κυάμων καὶ ἀρωμάτων.----ΟΓ, also the role of beans in the Greek
cult of the dead: Plut. Quaest. Rom. 95, Lydus Mens. 4.42; in Rome: Plin. HN 18.118.
The Flamen Dialis was forbidden beans (C. Koch, Der rémische Juppiter (Frankfurt,
1937] 34ff, 84f). On the Lemuria, above, n. 128, and Delatte, Faba 38ff. On the Indians,
L. von Schréder, Wiener Zs. ἡ Kunde d. Morgent. τς (1901) 187-212; on the Germans,
ibid., and M. Hifler, ARW 2 (1809) 109. On Ht. 2.17, see above, ch. ΠΕ, ἢ. 48.
4. Acusmata
acquitted, once more, of pure invention. In any case, the Pythagorean
taboos are closely connected with ritual, either taken over from it or
sct Up In Opposition to it.
Side by side with the ritual material, the acusmata have rules and
precepts that we would like to classify as rational, in the categories of
cthics!®? or physics. But the remarkable thing is the juxtaposition:
the rainbow is a reflection of the sun, thunder is a noise to frighten
souls in Tartarus, an earthquake is a mass meeting of the dead. Most just
is sacrificing, wisest is number, strongest is intelligence, holiest a
tallow leaf—taboo and proverbial wisdom jumbled together. Close
cxamination shows an amazing, inextricable tangle of religious and
rational ethics. A husband must not “persecute”’ his wife, for he took
her under his protection at the sacred altar; one should only help his
fellow to load up, never to unload, for we are in this life to be punished;
yood counsel is sacred—a moral commandment in religious form, like
the exhortation to leave children behind for the sake of the gods. The
command relating to behavior in battle, too (to fall, if one must, with
wounds in the chest) probably has a religious background (see n. 49).
It is striking how constantly attention is oriented toward the world
of the dead, the heroes, and the δαίμονες. Earthquake and thunder,
the clang of bronze and the ringing in one’s ear, the crumbs that fall
from the table—a person is always being moved and surrounded, even
physically, by “stronger” powers. In the saying that the motes in the
sunbeam are “souls,”!%8 this feeling is expressed in an almost discon-
certing manner; wherever a sunbeam falls, it is swarming with souls.
Another dictum that hints at “daemonic’’ forces is that attributed by
Aristotle to Philolaus: εἶναί τινας λόγους κρείττους ἡμῶν.139 The mallow,
the “holiest” thing, is a plant of the dead,!*° the λεωφόροι ατε the roads
over which the dead are conveyed to their graves! (and this is doubtless
also the reason why one may not split wood in the road),!4* and the
τρίγλη was sacrificed to Hecate.148 Pythagorean silence also belongs in
this context; the Greeks went past ἡρῷα in silence, in order not to disturb
the xpetrroves.1“4 In leaving home on a trip, one must not turn to look
137 Boehm, nos. 61-69 (‘‘Praecepta moralia’’).
138 Arist. De an. 404a17.
139 FE 1225230 = DK 44B16.
140 Nilsson, Op. I 336f, GrR I 705 n. 1.
141 Boehm on nos. 41-42.
142 No. 42 Bochm.
443 Apollodorus Περὶ θεῶν, FGrHist 244F109 = Ath. 7.325a-b.
144 Epicharm. fr. 165 Kaibel: ἀλλὰ καὶ σιγῆν ἀγαθόν, ὅκκα παρέωντι xappoves; cf.
Hsch. s.v. κρείττονες, Schol. Ar, Av. 1490.
I. PYTHAGORAS IN THE EARLIEST TRADITION
back “for the Erinyes are following.” The prohibition of bathing
may also be related to demonology.4® No wonder that we have
anecdotes pointing up how natural it seemed to the Pythagoreans to
encounter spirits!#4” Aristotle is our witness that the Pythagoreans
“used to express great surprise if someone said he had never seen a
daimon.’’148
The difference of the levels on which this “thought”’ operated does
not seem to have been recognized; and here we see the continuation
of a type of thinking that had already been left behind, in principle,
by Anaximander and Anaximenes. Both of these sought to explain the
earthquake on the basis of physical principles,!4® and thunder as well.15°
Recognition of the nature of the rainbow goes back to Anaximenes.!5!
But, whereas Anaximander asks, ‘‘ What is the sun?... How big is
it?’ and answers, “A circle of fire... the same size as the earth,’”452
the Pythagoreans ask, “What are the Isles of the Blest?’’ and are
satisfied with the answer, “Sun and moon.”
The intimations of number theory are also part of this picture.
“The wisest thing is number” ;18 such a sentence can be understood
without any esoteric doctrine and without any “Pythagorean mathe-
matics.’’ More important is the tetractys, regarded as the epitome of
Pythagorean wisdom. The Pythagoreans swore by Pythagoras as by
‘him who brought the tetractys to our generation,”
> A 4 ς f ~
οὐ μὰ TOV ἁμετέρᾳ γενεᾷ παραδόντα TETpAKTUV,
‘ 3 7 {
παγὰν ἀενάου φύσεως ῥίζωμά τ᾽ ἔχουσαν 154
The second verse of this couplet can scarcely be older than Empedo-
cles.155 Possibly the first line stood by itself in the beginning; with its
145 Above, n. 58.
146 Abstention from baths and from sexual intercourse is important not only in cult
but also in magic; see A. Abt, Die Apologie des Apuleius von Madaura und die antike Zauberei
(Giessen, 1908) 111ff, 114 nn. 5-6; C. Bonner, “‘Demons of the Bath,” in Studies Presented
to F. L. Griffith (London, 1932) 203-208.
147 Pythagoras and Calliphon: Hermippus ap. Joseph. Ap. 1.164; Eurytus and Philolaus:
Iam. VP 148, 139.
148 Fr, 193 = Apul. De deo δοῦν. 20.167.
149 Anaximander A28, Anaximenes A7§8, A2r.
150 Anaximander A23, Anaximenes A17.
151 Ay, A18; above, n. 30.
152 Anaximander AIT, 21.
163 Cf. above, n. 22; Aesch. Prom. 459, ‘Trag. adesp. 470 N., Gorg. Pal. 30; also below,
ch. VI. 1; VI 3.
154 References above, ch. 13, αν 120; Delatte, Litt, agg
™ Kranz, Philologus 1938, 438, ῥιζώματα B61, πηγή 23.10. The general idea of
φύσις is scarcely likely to have existed before the second half of the sth century.
4. Acusmata
negative formulation, the oath probably applied primarily to the
secrecy of Pythagorean doctrine. The τετρακτύς, a “tetrad’” made up
of unequal members, is a cryptic formula, only comprehensible to the
initiated. The word inevitably reminds of τρικτύς, the “triad” of
different sacrificial animals. Is the sacrificial art of the seer, involving
the shedding of blood, superseded by a ‘higher,’ bloodless secret ?1°*
‘The acusmata provide a hint toward an explanation: “What is the
oracle of Delphi?” “‘The tetractys; that is, the harmony in which the
Sirens sing” (lam. VP 85). The later tradition is more explicit: The
“tetrad” of the numbers 1, 2, 3, and 4, which add up to τὸ (the “ per-
fect triangle”), contains within itself at the same time the harmonic
ratios of fourth, fifth, and octave. The Sirens produce the music of
the spheres, the whole universe is harmony and number, ἀριθμῷ δέ τε
πάντ᾽ ἐπέοικεν 157 The tetractys has within it the secret of the world; and
in this manner we can also understand the connection with Delphi,
the seat of the highest and most secret wisdom.15* Perhaps Pythagorean
speculation touched upon that focal point, or embodiment, of Delphic
wisdom, the bronze tripod of Apollo. Later sources speak of its
mysterious ringing,®® which must have been ‘“daemonic”’ for Pytha-
yoreans.'6°
The meaning of the shibboleth or symbolon “tetractys” can only be
explained in a tentative fashion. In place of that which was connected
with it from the beginning, in the form of belief or experience, the
later sources give us more and more rationalizations. Some way or
186 Cf. Iam. VP 93, 147: Pythagoras taught Abaris to use a bloodless form of divination,
with numbers.
157 Connected in the tradition with the oath by the tetractys; above, ch. I 3, n. 126.
158 Boyancé, AC 1951, 421ff, cites the theory of the Delphians according to Plut. Quaest.
conv. 7458 (cf. De fato 568e), which connects the three Muses with three heavenly realms.
Cf. Heinze 75ff, Dérrie Hermes 1954, 336ff—Delatte, Litt. 260f, refers to the Delphic
KednSdves, who were brought into connection with the Sirens (Pi. Paean 8.71; Snell,
Hermes 90 [1962] 4f, Ath. 7.290e, Paus. 10.§.12).
189 The oldest testimony seems to be that of Vergil Aen. 3.92. Cf. K. Schwendemann,
Jb. 4. Dt. Arch. Inst. 36 (1921) 168ff. On the role of the tripod, see also P. Amandry,
La mantique apollinienne ἃ Delphes (Paris, 1950) 140ff, H. W. Parke and Ὁ. E. W. Wormell,
The Delphic Oracle (Oxford, 1956) I 24ff.
160 Eustath. p. 1067.59: καὶ of Πυθαγορικοί φασι τὸν χαλκὸν παντὶ συνηχεῖν θειοτέρῳ
πνεύματι" διὸ καὶ τῷ ᾿Απόλλωνι τρίπους τοιοῦτος ἀνάκειται. Cf. above, n. 34. For the
ringing of the tripods at Dodona, see Demon FGrHist 327F20. Hsch. s.v. Τρίοψ' 6 ὑπὸ
τῶν Πυθαγορικῶν ἐν Δελφοῖς τρίπους (cf. Por. VP 16) shows that Pythagoreans were
concerned with the Delphic tripod. The statement of Aristoxenus (fr. 15) that Pythagoras
got most of his doctrines from Themistocleia the Pythian priestess, could well be a
rationalizing interpretation of the connection suggested by the acusina about the tetractys.
For speculation about the relation of the tripod to the number 3, see Plut. De FE 1876,
Lobeck 386f. |
II, PYTHAGORAS IN THE EARLIEST TRADITION
other the secret of the world is to be found in number, but there is
danger in too much conjectural reconstruction. If this is a starting
point for Pythagorean science, the question remains, how far Pytha-
goras went in this direction. The pronouncement that number is
“τῆς wisest thing” may be made on no other basis than naive wonder-
ment at its versatile usefulness and its indefeasible correctness, with
no foundation of sophisticated number theory or mathematical
philosophy. It does not take more than the amusement of an idle hour
to discover and establish that 1+2+3+4=10. Numerical relationships
in the cosmic order are to be found in primitive and mythical thought,
and in the same realm we can find the notion of the cosmic origin
and function of music, and it is not necessary to assume special astro-
nomical knowledge. The evidence about the tetractys, about num-
bers, and about music, is not necessarily on a different level from that
about other acusmata, such as that an earthquake is a rally of the dead,
that the rainbow is a reflection of the sun, or that the sun and moon are
the Isles of the Blest. To what extent the seeds thus planted had already
been developed in the direction of rational science and knowledge,
is something which the history of the exact sciences must try to
clarify.
The question of the date of this Pythagorean wisdom, thus far
postponed, cannot be answered in a completely satisfactory manner.
The latest terminus for the form of those that are sayings or precepts
without explanation is the allegorical treatment of Anaximander the
younger, about 400 B.c., and for the existence of a sizable, varied
collection it is the reports of Aristotle; but in the other direction there
can scarcely be any limit. In reckoning probabilities one must bear in
mind that in oral tradition such a loose aggregation of adages and
maxims is liable to constant change; some items are lost, but others
are bound to take their place, especially when there begins to be a
certain competition to know “‘as many as possible” of them (Iam.
VP 82), and by the rule of αὐτὸς ἔφα all are attributed to the Master.
Still we may be confident, in the light of the zeal to maintain the
doctrine of the Master himself, that at least some original matter is
faithfully transmitted. It is like a gravel pile; there is no pebble of
which we can say that it must be primitive rock, but any single one may
be. |
We can get a little beyond this non liquet by recognizing that not
only the content, in certain details, but certainly the form, as a whole,
is older than Pythagoras, There are taboo-precepts in all primitive
4. Acusmata
cultures; and the aphoristic formulation is attested for the Seven Sages.
Individual rules, and even the number symbolism} and the assertion
about the rainbow,!® are attested for an earlier period. And it is
unthinkable that ritual prohibitions like those of beans, heart, and
baths, widespread as they are, and variously modified, can have had
their origin in the doctrine of the historical Pythagoras. As befits the
role of hierophant which Pythagoras played according to the legend,
Pythagoreanism is attached to preexisting Greek cults.
The idea occasionally expressed,!® that the acusmata were simplistic
back formations from a developed, scientific doctrine of Pythagoras, is
thus in principle refuted. It is true that the history of human thought
shows relapses from physical science to magic, but vestiges of
science tend to be preserved. Insofar as Pythagoreanism agrees with
more ancient material in general Greek cults, it is only by petitio
principii that we could imagine a double, self-cancelling development
—forward to science and back again to the starting point.
The only remaining question would be, then, whether the acusmata
can have worked their way into the tradition from outside, at a late
stage in its development. It is not impossible that this may be the case
with a few individual items, but for the entire collection it is utterly
improbable. There must have been a point of crystallization for any
possible secondary additions. In fact, even the later “mathematical”’
Pythagoreans conceded that the acusmata came from Pythagoras.
Thus the modern student too may, or rather must, conclude that a
nucleus of the collection goes back to Pythagoras and that the doctrine
of the historical Pythagoras was presented on the level of the acusmata
and passed on in this form.
For the pronouncements of the acusmata fit in with what we can
learn from the Pythagoras legend: the same universe of gods, daemons,
and souls interpenetrates the universe of man; we hear of Pythagoras
as divine; we are reminded of metempsychosis; and all of this has
its source in a “wisdom” that comprehends equally and without
161. Above, nn. 47, 84.
162 Below, ch. VI 3. .
163 Providing that Anaximenes (A7, 18) is earlier than Pythagoras, as in the generally
accepted chronology.
164 Krische 33ff, and more recently Maddalena 362 (above, ἢ. 82). .
165 For example, astrology can be interpreted as a retrogressive development of Hellenis-
tic science (see Nilsson IH? 268). But it preserved, from its scientific background, the order
of the planets, the spherical shape of the earth, and advanced methods of calculation.
166 Below, ch. Il 5; ἐκάλει (sc. Pythagoras), Arist. fr. τού,
Il. PYTHAGORAS IN THE EARLIEST TRADITION
differentiation the divine and the carthly, the rational and the religious—
the lore of one who “knows more”’ than ordinary men.
Even if Pythagoras was adapting older material, there was naturally,
in the process of choice, a certain λόγος, a kind of “reflection;” but
this does not imply a rationally constructed system. Decisions may be
taken in quite different strata of the psyche, as to what is appealing, or
illuminating, or obvious. It is questionable to what extent one is
justified in speaking of a “purification” or a “‘spiritualization” of
religion.1®’ There is, in the form and function of the acusmata, a forward
step; but this is on a different level.
The ritual prescriptions which made their way into the acusmata
serve for unusual occasions. Their demands apply during the initiation
ceremony, or incubation, or a journey to the underworld, or for the
period of convalescence; afterward, one can live as he did before. The
rhythmic alternation of holiday and nonholiday, of ἱεραί and ὅσιαι
ἡμέραι, is a mark of all primitive, naive religion. In such religion
opposites dwell peaceably side by side—eddynpia and αἰσχρολογία,
chastity and license, earnest and jest, each in its proper time. The
““sacred’”’ animal is sacred just because one day it will be slaughtered
and eaten. Vegetarianism and omophagia are not mutually exclusive.
These rhythms respond to basic needs of the human psyche, and of
society. But, when reason begins to reflect on man’s beliefs and
behavior and tries to remove discrepancies and to arrive at general,
objective theses about gods and their laws, the ancient way of living
must be reformed; whatever gods prescribe must be valid at any time.
If Pythagoras himself was a kind of hierophant, he found no suc-
cessor ;168 the Pythagoreans were left with their acusmata applying no
longer to festivals but to normal life, which, as a consequence, seemed
to others abnormal. Prohibitions like those of beans, heart, certain
fishes, and baths are now absolute and must be observed at all times;
and the Pythagorean always wears white clothing. He lives every day
of his life as though he were preparing for initiation at Eleusis, for
incubation at Asclepius’ temple, or for the journcy to Trophonius. He
follows not the cult rules of a certain holy site, but those of a βίος
167 Zeller I 411: “(We shall scarcely go wrong in believing) ... that he transformed the
Dionysiac mysteries in the spirit of a more advanced moral teaching, and made them
into an auxiliary of it.” Perhaps the aeusma about salt can be interpreted in this way
(above, n. §2).
"8 Later tradition constructed a list of “scholarchs,”” but there ts not even agreement
among the sources on the tmamediate successors of Pythagoras, Cf ch. ΕΠ 2. ἢν 33.
4. Acusmata
which he has personally and consciously chosen.1®° To be sure, as is
clear from the example of vegetarianism, an open breach with ritual
picty is avoided.
As metempsychosis changed from ritual and myth to a doctrine
witha claim to truth, so here, ritual bound to certain conditions changed
into unconditional, permanent rules of life. In both cases Orphism, or
the ᾿Ορφικὸς Bios, bears an embarrassing resemblance to Pythagoreanism.
In Orphism, however, according to the testimony of Plato, the older
practice of individual, magic rites did not die out; he makes the com-
plaint against the Orphic ἀγύρται καὶ μάντεις that they promised
individuals and whole cities expiation for their sins, at the cost of a
little sacrifice and a pleasant dinner.1’° One can make use of the ᾽Ορφικά
without being an Orphic, but he who follows Pythagoras becomes
a Πυθαγόρειος.
To take the acusmata seriously means an almost frightening constric-
tion of one’s freedom of action in daily life. Whether a Pythagorean
vets up or goes to bed, puts on his shoes or cuts his nails, stirs the fire,
puts on the pot, or eats, he always has a commandment to heed. He
is always on trial and always in danger of doing something wrong. No
more carefree irresponsibility! Everything he does is done consciously,
almost anxiously. The mythical expression of this attitude to life is a
world full of souls and daemons, which affect every moment of a
person’s life. Everywhere are rules, regulations, and an ascetic zeal for
discipline; life is πόνος, which must be endured.
In his discussion of Greek shamanism, Dodds uses the word
“puritanism,”?”! and by it he means the strongly felt tension between
bodily needs and those of the soul, which is to be freed from the body.
If our analysis is correct, the comparison with historical Puritanism
can be seen in a still broader perspective. The Puritan and Pietist
movements emerged as reactions against a Christianity relying on
“primitive” religious forms, in which ritual and daily life, Church
and world, holiday and workday ran along parallel. Their aim was to
make the whole of life a service of God; every day was to be lived like
Good Friday. The dangerous area of arbitrary human choice and of
9 Cf. Pl. Rep. 6oob. If Iam, VP 96-100 is mainly from Aristoxenus, it provides a
very early report of a communal society of the monastic type.
πὸ Rep, 364b-c.
71 Jer, 139(ff). A comparison has also been drawn, occasionally, between the Pythago-
rean life and that of Calvin's Geneva.--On “tighicher Gottesdiense” see Nilsson TP
δα WH, Gomperz (ΡΠ δὲ sof) has good remarks on the “acusmatic” life.
II, PYTHAGORAS IN THE EARLIEST TRADITION
carefree joy in living was narrowed as much as possible. The ecclesi-
astical hierarchy was replaced by the democratic conventicle.
| Thus the acusmata go beyond ritual piety, not in their content but
in the way in which they regulate man’s life, and foreshadow a later
development in Greek ethics, much in the way that the doctrine of
metempsychosis foreshadows a later ontology. The significance of
Pythagoras is to be sought not in the realm of philosophy proper but
in the approaches to it, in his position as an intermediary between old
and new.
§. ACUSMATICI AND MATHEMATICI
The tradition has contradictory reports about sects, hierarchical
gradations, and schisms among the Pythagoreans. Very often a
distinction is made between a lower and a higher degree of Pytha-
gorean wisdom, and this goes back at least as far as Timaeus.! He tells
᾿ a five-year period of probation, during which the new disciple must
isten in silence, and di
τ , and did not even see Pythagoras face to face; the voice
of the sage came from behind a curtain (σινδών). Only after this period
did ag pupil become ἐσωτερικός. The distinction of Πυθαγορισταί and
Πυθαγόρειοι is joined to that of “exoteric”’ and “esoteric,’? and there
is also a distinction between ἀκουσματικοί and μαθηματικοί.3 The
mathematici are the “genuine,” or truly “phi izing”
u hilosophi
eas τ : y “p phizing”” members,
goal 15 ἀκρίβεια, so that they correspond to the “esoteric”
Pythagorcans.4 T iti 1 ἱ
᾿ g he position of the πολιτικοί is confused; sometimes
they are reckoned among the acusmatici, sometimes among those of
: on a.
the highest grade.> There are some apparently artificial divisions into
: : : ᾿
sea §5§6F13 = D.L. 8.10 is very like Iam. VP 72(ff), so that the latter’s exposition
as a whole must be based ultimately on Timaeus (Rostagni, SerMin II 1.21ff, von Fritz
Pol. 39, parallel V). See also the collection of references in Zeller I 400 n. 4, and von Frits:
Aa 1960.—The king of Persia eats behind a curtain, according to Ath. ne as
Hippol. Ref. 1.2.4 (τελεώτερα οὐ μετριώτερα μαθήματα); Iam. VP 80 (Πυθαγόρειοι
as Mie like *Arrixoi distinguished from ’Arrixtorai—an interesting point chronolo-
ες he a geile math. sc. 24 PP 74.15~75.§ (οἱ ἔξω... of ὀνομασθέντες Πυθαγόρειοι)
οἱ, eocr. 14.§ on the difference between respectably livi }
and beggar Πυθαγορισταί. Pm dM Seen renee
᾿ Por. VP 37 (characteristic of the mathematici are ἀκρίβεια, περιττότερος λόγος); Clem
᾿ . Strom. 5-59 (γνησίως); lam. VP 20 (φιλοσοφοῦντες distinguished from the πολλοί;
this digression, which breaks into the continuity of the sentence, is athetized by Picubrien,
SBBIn 1935, 663f; but Iamblichus is capable of such a thing).
“As also in Inamblichus’ superficial summary, §89.
Stam. VP 1$0: ἀκουσματικοὶ καὶ πολιτικοί dis ingui
40: stinguished from θεωρητικοὶ καὶ φιλόσοφοι
Matt HOMIE! ap. August, Ord, 2.20 presents the πολιτεκοί as the highest group : cf,
am Vl 72, rok, tay, Phe demands of practical life, shown in’ section 88, are incom-
patible with Plato's idealistic demands,
5. Acusmatici and Mathematici
three groups, which scem likely to be secondary.* The main point is
always the existence of differences of rank or grade, but they are
sometimes seen as steps in a continuous development and sometimes
as names of separate groups. In this context the word ἀκουσματικοί
remains a puzzle. If the reference is simply to the probationary period
of “hearing,” one would expect ἀκουστικοί or ἀκροατικοί. What
acusmata meant to the Pythagoreans we are not told in this connection.
Quite different is a notice that Iamblichus copied twice from the
same source, not without introducing a serious contradiction on one
point. There were, he says, two kinds of Pythagorean philosophy, as
there werealso two kinds of Pythagoreans, ἀκουσματικοί and pabnparucot.
In what follows, the text of the De communi mathematica scientia has
preserved the original version:®
8 ἀκουστικοί, μαθηματικοί, φυσικοί: Taurus ap. Gell. 1.9; epigram of Socrates (cf.
D.L. 2.47, Carcopino, Bas. 2$4.2-3: conjectural date, Ist century A.D.), A. P. 14.1; cf.
the Stoic division of philosophy into the three parts logic, physics, and ethics.—oeBaorixot,
πολιτικοί, μαθηματικοί: Anon. Phot. 438b19ff, Schol. Theocr. 14.5 (σεβαστικοί are
defined as of περὶ τὴν θεωρίαν).--Πυθαγορικοί-- Πυθαγόρειοι--Πυθαγορισταί as immediate
pupils—pupils of pupils—egwAer ζηλωταί: Anon. Phot. 438b23ff, Schol. Theocr. 14.5.
? ἀκουστικοί, above, n. 6. ἀκροατικοί = ἐσωτερικοί used of the pupils of Aristotle, Gell.
20.§ (on this tradition about the Peripatos see Lucian V. auct. 26, Plut. Alex. 7, G. Boas,
Philos. Rev. 62 [1953] 79ff).
8 Iam. VP 81, 87-89. (The account of the acusmata, which comes mainly from Aristotle,
comes between these two passages, from 82-87, i.e. pp. 47.4-$1.12 Deubner. On this,
see above, ch. Π 4. How mechanically the insertion is made is shown by the word τούτους,
p. 51.13, which has as antecedent the ἀκουσματικοί named at pp. 46.25 and 47.1. The
scissors-and-paste procedure is surely lamblichus’ own.) This passage is nearly the same
as Comm. math. sc. 25, pp. 76.16-78.8. The latter passage was written later, and has the
error ἐξαγώνων for πενταγώνων at p. 77.20. It was not, however, copied from the Vita
pythagorica, for it continues without a break, whereas the VP shows arbitrary alterations.
To set the crucial sections parallel to each other:
Iam. VP 81 p. 46.26ff:
τουτωνὶ δὲ of μὲν μαθηματικοὶ ὦμολο-
γοῦντο Πυθαγόρειοι εἶναι ὑπὸ τῶν ἑτέρων,
τοὺς δὲ ἀκουσματικοὺς οὗτοι οὐχ ὧμολό-
γουν, οὔτε τὴν πραγματείαν αὐτῶν εἶναι
Πυθαγόρου, ἀλλ᾽ ἹΙππάσου" τὸν δὲ ἽἽππα-
σον of μὲν Κροτωνιάτην φασιν, οἱ δὲ Μετα-
ποντῖνον...
Comm. math. sc. p. 76.19ff:
, 4 4 4 > 4 t
τούτων δὲ of μὲν ἀκουσματικοὶ ὡμολο-
γοῦντο Πυθαγόρειοι εἶναι ὑπὸ τῶν ἑτέρων,
τοὺς δὲ μαθηματικοὺς οὗτοι οὐχ ὧμολό-
γουν, οὔτε τὴν πραγματείαν αὐτῶν εἶναι
fa 3 a € La 4 3}
Πυθαγόρου, ἀλλὰ Ἱππάσου" τὸν δ᾽ Ἵππα-
ε 4 , fa ε 4
σον οἱ μὲν Κροτωνιάτην φασίν, ot δὲ Μετα-
ποντῖον. οἱ δὲ περὶ τὰ μαθήματα τῶν
Iam. VP 87 p. 51.128: Πυθαγορείων τούτους τε ὁμολογοῦσιν
¢ 4 ᾿ 3 , ~ f f 4 + é ¥
of δὲ περὶ τὰ μαθήματα τῶν Πυθαγορείων εἶναι Πυθαγορείους, καὶ avrot φασιν ἔτι
i € “~ ~ 3 Δ rd + 1 > ~
τούτους τε ὁμολογοῦσιν εἶναι Πυθαγο- μᾶλλον, καὶ ἃ λέγουσιν avror ἀληθῆ
ρείους, καὶ αὐτοί φασιν ἔτι μᾶλλον, καὶ εἶναι...
ἃ λέγουσιν αὐτοί, ἀληθῆ εἶναι...
Nothing can be changed in VP 87, because in the following passage the interpretation
of the mathematici is given. Nauck and Hélk (5) proposed simply emending 81 after the
Comm. math. sc. passage, but this is impossible. See below, n. 9 (DK I 107 π.; Deubner,
SBBlIn τοῖς, 620): Ilamblichus’ source is correctly reproduced in Comm. math. sc. and this
is the only one with historical value. The acusmaticus Hippasus is a conjecture of Iam-
blichus. (See further Delatte, Litt, 272f; Prank, Logos 9 [1920-1921] 246 n. 1; Rey 228
n. 1: von Fritz, SBMii 1960, 21: Timpanaro Cardini has it wrong, 80.) The text used in
IK should therefore be that of the Comm, math, se.
I. PYTHAGORAS IN THE EARLIEST TRADITION
Of these, the acusmatici are recognized by the others as Pythagoreans,
but they do not recognize the mathematici, saying that their philo-
sophic activity stems not from Pythagoras but from Hippasus ...
But those of the Pythagoreans whose concern is with the μαθήματα
recognize that the others are Pythagoreans, and say that they them-
selves are even more so, and that what they say is true.
According to this, the acusmatici are incontestably Pythagoreans, who
refuse to recognize the mathematici,who, they say, are really followers
of the innovator Hippasus. On the other hand, the mathematici maintain
that they are the true successors of Pythagoras, “even more”’ than the
acusmatici, and that the alleged innovation of Hippasus was nothing
more than a plagiarism of doctrine taught much earlier by Pythagoras;
“everything is due to ‘that man’”’—Pythagoras,
In his De vita Pythagorica, lamblichus seeks to maintain just the
opposite of this: that the mathematici are the uncontested Pythagoreans,
who refuse to recognize the acusmatici on the ground that they are a
sect founded by Hippasus. Thus Hippasus is an acusmaticus in this
version, and in the other a mathematicus. But this is not maintained
consistently in the Vita Pythagorica; after a digression on the acusmata
follows the other version, agreeing with De communi mathematica
scientia. The latter, which is consistent and complete in itself, is thus
shown to be primary. The cause of the discrepancy is not a slip of the
pen, for in two other passages Iamblichus calls Hippasus an acusmaticus.®
Actually, the account designed for the Vita Pythagorica is the only one
conceivable to Iamblichus. For him the μαθήματα, the Wissenschaft,
the incontrovertible proofs—so incontrovertible that it is not worth-
while to think them through again—!® belong irrevocably to the
doctrine of Pythagoras. It seemed to him unthinkable that anyone
could contest this, to say nothing of these doubters being acknowledged
by their opponents as genuine Pythagoreans. Iamblichus knows the
tradition that made the acusmatici the lower class, the “spurious,” the
“many’’ who are not true philosophers. Here he can only believe that
his eyes have deceived him, and quickly switch the two nouns." We
®Tam. In Nic. 10.20 (DK 18.11), Jam. De anima ap. Stob. 1.49.32. Syrianus derives his
material from Iamblichus (Met. 123.7ff, 142.21ff, with the same apocryphal citation of
Hippasus). Thus Jamblichus is the only source for the tradition of Hippasus as an acus-
maticus.
Cf. Iam. VP 157: τὰ γραφέντα ὑπὸ τῶν Πυθαγορείων ὑπομνήματα, περὶ πάντων
ἔχοντα τὴν ἀλήθειαν. ., καὶ πραγμάτων ἐναργῶν καὶ ἀναμφιλέκτων ὡς ὅτι μάλιστα μεστὰ
μετὰ ἀποδείξεως ἐπιστημονικῆς καὶ πλήρους, τὸ λεγόμενον, συλλογισμοῦ...
Y Atiam. VP 871 in the version of the mathematici, this exchange produces an impossible
result, and Tumblichus should have dropped the whole text. But it suited his predilections
so well (ef FP do, goth) chat he could not let it go.
5. Acusmatici and Mathematici
have here, then, an arbitrary alteration, whose motive is transparent;
but it is not maintained consistently, and the result is confusion.
The only version usable in an attempt to reconstruct the history of
carly Pythagoreanism is that of the De communi mathematica scientia.
It presents us with two rival groups of Pythagoreans: the acusmatici
gct along without proofs—and in general without the branches of
learning that fall into the special category of μαθήματα, namely
arithmetic, geometry, astronomy, and music (as we may discover from
the rest of the tradition), whereas it is in these that the mathematici see
the true, more profound wisdom of Pythagoras. What is left to the
acusmatici is a collection of dicta without explanation—the acusmata,
and this is surely the source of their name. Iamblichus is doubtless
justified by the content, though it makes awkward reading, in inserting
here the section on the acusmata. The mathematical and scientific
aspect is taken from Pythagoras and attributed to Hippasus, as a later
development which turned the school tradition aside from its original
course. The mathematici have a different explanation:
Pythagoras, they say, came from Ionia and Samos at the time of the
tyranny of Polycrates, when the civilization of Italy was flourishing,
and the first men in the cities became his trusted associates. The older
of these he addressed in simple style, since they had little leisure,
being occupied with political affairs, and he saw that it was difficult
to speak to them in terms of μαθήματα and proofs. He thought they
would be better off for knowing how to act, even without knowing
the reasons, just as persons under medical care get well even though
they are not told the reason for every detail of their treatment. The
younger men, however, who had time to put in the effort of learning,
he addressed with proofs and μαθήματα. They themselves, then, the
mathematici, are the successors of the latter group, and the acusmatici
of the former.
Hippasus, they said, only published, for his own aggrandizement,
things that Pythagoras had taught long before.
When we approach the problem of the source, and therefore also
the value, of this report, we are struck immediately with the imparti-
ality with which these two contradictory versions are presented; the
author docs not start with a preconceived answer. In each of the other
reports of lower and higher levels of Pythagorean wisdom, only one
side is given; they follow the version of the mathematici, who distin-
guish themselves, as “more genuine” Pythagoreans, from others who
Π. PYTHAGORAS IN THE EARLIEST TRADITION
profess a simpler form of Pythagorean doctrine. From the time of
Timaeus, the account of the mathematici holds the field, and all that
remains of the other side is the unexplained term acusmatici. Iamblichus’
account of the schism, which gives the other side a chance to be heard
along with, and even before the mathematici, cannot possibly be derived
from the one-sided version that prevailed from Timaeus’ day,}° and
therefore must be, in essentials, earlier than Timaeus.
Now the term acusmata is only found in the exposition of Iamblichus,
which goes back to Aristotle, while the non-Aristotelian tradition uses
the word symbola. This suggests that the report of the schism also stems
from Aristotle, and that Iamblichus has, in a way quite consistent
with his usual compilatory method of writing, artificially rearranged
two excerpts that come, ultimately, from the same source. This
conjecture is strongly corroborated by the introductory sentence in
De communi mathematica scientia: δύο δ᾽ ἐστὶ τῆς ᾿Ιταλικῆς φιλοσοφίας
εἴδη καλουμένης δὲ Πυθαγορικῆς (p. 76.168). This corresponds very closely,
without being a direct copy, to expressions in Aristotle, though
elsewhere, understandably enough, Iamblichus does not use the
expression καλούμενοι Πυθαγόρειοι. In addition, the language of the
passage that follows in De communi mathematica scientia shows close
kinship, without being a quotation, with expressions in the Meta-
physics2® Thus we have a piece of evidence traceable to Aristotle,
which in its very wording has preserved marks of its origin.1”
Iamblichus’ account of the division among the Pythagoreans,
unique in content, takes on tremendous significance because of the
12 Notice the expression ἔτε μᾶλλον and the emphasis on ἀποδείξεις at Iam. VP 87
taken along with the expressions collected in nn. 2-6 above.
18 This is perceived by Hdlk 5f.
14. This solution is favored by Delatte, Litt. 273(ff), though he too readily takes the passage
on the acusmata as a unity with the surrounding material about the division of the society
(cf. ch. II 4). Bertermann assigned the whole to Androcydes, because in his superficial
way he failed completely to notice the difference between the conception of the acusmata
held by Aristotle from that of Androcydes.
= Mete. 342b30: τῶν ᾿Ιταλικῶν τινες καὶ καλουμένων Πυθαγορείων; Cael. 293420:
οἱ περὶ τὴν ᾿Ιταλίαν, καλούμενοι δὲ Πυθαγόρειοι. The adjective Πυθαγορικός is found
at De an. 407b22.
16 Above, ch. I 2, n. 112.
7 It is hard to say whether some material from other sources may be interspersed with
the Aristotclian. {t is perhaps doubtful whether Iam. VP 86 (pp. §0.18-51.6, and 41.7--12)
or vp 89 = Comm. math. sc. pp. 77.24-78.5 (discussed below in ch. VI 3) come from
Aristotle. The concluding sentence, Comm, math. sc. p. 78.6, is Aristotelian in its phrasing
(περὶ μὲν οὖν τῆς διαφορᾶς ἑκατέραν τῆς πραγματείας καὶ περὶ τῶν μαθημάτων σχεδὸν
ταῦτά τε καὶ τοιαῦτά ἐστι τὰ συμβεβηκότα). !
5. Acusmatici and Mathematici
authority of its source.!® In fact, the modern controversies over
Pythagoras and Pythagoreanism are basically nothing more than the
continuation of the ancient quarrel between acusmatici and mathematici.
Is there nothing more in the doctrine of Pythagoras than what is
indicated by the acusmata, with which the Pythagoras legend and
the theory of metempsychosis are of course closely connected? Or
was there from the beginning, behind these religious and mythical
features, whose existence cannot be denied by the modern scholar
any more than it could by the mathematici, a new, scientific ap-
proach to philosophy, mathematics, and the study of the world’s
nature?
Before pronouncing on this question,‘ we must discuss the report
in the light of other testimony from fourth-century sources, so as to
confirm its Aristotelian origin and at the same time get from reports
to historical facts. Aristotle recognizes among the Pythagoreans a
twofold πραγματεία: on the one hand Πυθαγορικοὶ μῦθοι, metem-
psychosis, the Pythagoras legend, and the acusmata, and on the other
a philosophy of number connected with mathematics, astronomy, and
music, which he never tries to trace back to Pythagoras himself and
whose chronology he leaves in abeyance. In this he dissociates
himself from the Platonists, who attribute to Pythagoras himself a
more sophisticated version of the Pythagorean number theory.”
Plato and his pupils thus stand in the tradition of the mathematici, and
it is not surprising that their version carried the day. The doxographical
tradition, too, was dominated by the Platonic interpretation of
Pythagoreanism.
18 The account of the division into groups is taken seriously by Rohde, Q 107ff; Holk
sf; Delatte, Litt. 29ff, 271ff, Pol. 25ff; Jaeger, SBBln 1928, 416 = “On the Origin and
Development of the Philosophic Ideal of Life,” tr. R. Robinson, in Aristotle, 2nd Eng.
ed. (Oxford, 1948), 455ff; Frank 69ff; Burnet, EGP 94; Minar 34f; Guthrie I 192f; and
others. On the other hand, Zeller (I 415 n. 1) rejected the whole tradition as a late con-
struct. It would not be unthinkable that Aristotle, having observed differences and worked
out the idea of a split in a theoretical way, then came to think of this as a historical reality;
but the explicit statements as to the extent to which each group recognized the other
cannot be accounted for in this way.
19 Frank (71) finds the exposition of the mathematici simply unsinnig. Speaking for the
acusmatici as the original group are Rohde, Q tozff, Holk 4, (with reserve) Delatte,
Litt. 311f; on the other side, among others, is Jaeger (Paideia I 221 = Eng. ed. I 162).
Even von Fritz (5 ΒΜὴ 1960, 14, 18ff) is overborne by the weight of the later tradition and
secks to weaken the force of the report. The acusmatici, he thinks, were only concerned
to reject certain developments of doctrine, whereas the mathematici wished to emphasize
the “truth” of their findings and did not attribute these to Pythagoras. But the words of
the text are unambiguous: εἶναι δὲ πάντα ἐκείνου τοῦ ἀνδρός (Iam. VP 88). Guthrie
seems to forget the claim of the acusmatici (1 192).
© Cf above, ch. I 2, 3, and 4.
I, PYTHAGORAS IN THE EARLIEST TRADITION
We can identify certain Pythagoreans who may be classed among the
mathematici: first of all Archytas with his pupils and also some of his
predecessors, among whom he himself mentions the name of Eurytus;
doubtless also Simmias and Cebes, the “auditors’’ of Philolaus; and
finally Echecrates of Phlius,? with his friends. It is this same circle to
whom Aristoxenus refers:
For the last of the Pythagoreans, whom Aristoxenus himself saw,
were Xenophilus of Chalcidice in Thrace, Phanton of Phlius, and
Echecrates, Diocles, and Polymnastus, also Phliasians. They were
pupils of Philolaus and Eurytus of Tarentum.”
Aristoxenus named Archytas as a friend of his father Spintharus.?®
Whatever its basis may have been, Aristoxenus’ assertion that
Pythagoras was especially fond of beans obviously has a polemical
point; and it is now clear that it is aimed at acusmatici. Equally clearly,
his statements about Pythagoras eating meat are to be explained along
the same lines;?4 and perhaps the main purpose of the “Πυθαγορικαὶ
ἀποφάσεις was to drive out of currency the enigmatic or ridiculous
acusmata.
When we start looking for acusmatici, we think first of the
Πυθαγορισταί35 who appear in leading or secondary roles in the Middle
21 Above, ch. I 4, n. 40. A woman from Phlius belonged to the Academy (D.L. 4.2).
22 Aristox. fr. 19 = D.L. 8.46; the same names are given in fr. 18 = Jam. VP 251.
23 Above, ch. II 1, n. 50.
24 Above, ch. II 4. Aristoxenus gives the same historical account as the mathematici:
Pythagoras comes to Italy in the time of Polycrates (fr. 16), and acquires influence over the
political leaders of the city (frr. 43, 17, 18).
25 Cf. Zeller ΠῚ 2.93f, Weiher 55-68, Méautis 1off. The most important texts, from
D.L. 8.37 and Ath. 4.160f, are printed in DK 58E. The fragments in question are the
following (Kock’s numbers, which are followed by Edmonds): Antiphanes Corycus fr.
13§ = Ath. 4.161a, Mnemata fr. 160 = Ath. 4.161a, Neottis fr. 168 = Ath. 4.108e-f
(above, ch. II 3, ἢ. 106), inc. fab. fr. 226 = Ath. 2.60d; Mnesimachus Alcmeon fr. 1 =
D.L. 8.37; Alexis Pythagorizusa fr. 196-197 = Ath. 4.161c-d, fr. 198 = Ath. 3.122f,
fr. 199 = Gell 4.11.8, Tarentini fr. 219 = Ath. 11.463d—e (see Weiher 66f), ἔστ. 220-221 =
Ath. 4.161b—c, fr. 222 = Ath. 4.134a-b; Aristophon Pythagoristes fr. 9 = Ath. 4.161e,
fr. τὸ = Ath. 6.238c-d (see Méautis 12ff), frr. 12-13 = D.L. 8.38; Cratinus the younger
Pythagorizusa fr.6 = D.L. 8.37, Tarentini ἔτ. 7 = ID.L. 8.37.—There is no way of knowing
whether Naevius’ Tarentilla belongs in this context.—Delatte conjectures (Vie 241) that
the plays of Alexis and Cratinus entitled ‘Varentini and those entitled Pythagorizusa were
identical, in each case, and that their authorship was contested; but here again there is no
basis for a firm answer.—The difference in the appellations Πυθαγοριστής (Aristophon:
title and frr. 9, 12), Πυθαγορικός (Antiphanes fr. 160), Πυθαγόρειος (Alexis fr. 196),
and Πυθαγορίζοντες (Alexis fr. 220) is not a matter of technical terminology (cf. Weiher
56). The dietary regulations are in any case based on the authority of Pythagoras himself
(Antiphanes fr. 168).
5. Acusmatici and Mathematici
Comedy. It is repeated again and again that they eat οὐδὲν ἔμψυχον, 3 ὃ
not even the meat of sacrificial animals,?”? only a lot of wretched
vegetables.28 Sometimes they scarcely eat anything at all;?® they drink
plain water,?° attract attention with their silence and their orvyvé7ys,!
they wear a ragged τρίβων,}5 go about barefoot,%% and are stiff with
dirt since it is against their principles to bathe.** Shabbiness turns into
arrogance; they are typical ἀλαζόνες. 55
The comic poets mock these mendicant Pythagoreans as living so
ascetically because this was the best they could do;** but a fragment of
Aristophon reveals another motive: “He said that he had gone down
to visit those below in their daily life, and he had seen all of them, and
that the Πυθαγορισταί had far the best lot among the dead. For Pluto
dined with them alone, because of their piety.’’? The interlocutor
then remarks that Pluto must be a very easy-going god, to associate
with such dirty ragamuffins; but in spite of the element of caricature,
the goal of Pythagorean asceticism is clearly apparent: a favored
life in the next world and intimate acquaintance with the gods.*
The reference to the ‘thrice blessed” Pythagoras, in Antiphanes,
shows that the “Pythagorists’’ were also concerned with metempsy-
chosis.??
These Pythagoreans are placed in the south Italian scene by the very
title, Tarentini. In a play of Alexis, who himself came from Thurii, the
happy-go-lucky life of Athens is described to one of the characters;
obviously the play contrasted the sobersides Pythagorean and the
casy-going Athenian.4° At the time of the play’s performance—
26 Antiphanes fr. 135, Alexis fr. 220, Mnesimachus fr. 1, and Alexis fr. 27 (Méautis
10 n. 6.)
27 Mnesimachus fr. 1, Alexis fr. 196.
28 θύμον, Aristophon fr. 10, Antiphanes fr. 168; ἅλιμα καὶ κακὰ τοιαῦτα, Antiphanes
tr. 160.
28 Aristophon fr. 10, Alexis frr. 221, 197, Theocr. 14.5ff.
30 Alexis frr. 198, 220, 221; Aristophon fr. 10.
31 Alexis fr. 197.
32 Aristophon frr. 9, 13.
83 Aristophon fr. 10, Theocr. 14.6.
84 Gdovoia, Alexis fr. 197, Aristophon frr. 13, 9, 12; lice: Aristophon fr. 13.
46 Com. adesp. fr. 275 Kock fits into this context: οὐδ᾽ ᾿Ιταλιώτης οὐδ᾽ ἀλαζὼν οὐδαμῶς.
46 Antiphanes fr. 226, Aristophon fr. 9.
” Aristophon fr. 12. Méautis (16) conjectures that the person mentioned is Pythagoras
himself; but can he be supposed to meet his followers in Hades?
48 Cf. Emp. fr. 147, Pl Phdr. 2474.
%” Antiphanes fr. 168; Aristophon fr, τὸ (cf. Weiher 65, Méautis 12ff) mocks palin-
genesis as an ἄλλοτε ἄλλον γίγνεσθαι (1).L. 8.36).
40 Alexis fr. 222.
II. PYTHAGORAS IN THE EARLIEST TRADITION
probably in the middle third of the fourth century*W—" Pythagorists δ
must have been a familiar sight in Athens.
Clearly, Aristoxenus’ veiled polemic has the purpose of destroying
this image of Pythagoreans as a group of low-class tatterdemalions,
addicted to superstitious abstinences. There is special point in his
insistence that the Pythagoreans he knew personally were “τῆς last’’5
‘They held to their original way of life (ἤθη), and their science ..,
until, not ignobly, they died out” (fr. 18 = Iam. VP 251). Nothing:
about abstinence from beans or meat—only μαθήματα; nothing.
about poverty, dirt, or lice—only noblesse, to the very end.” Diodorus,
in a passage certainly based on Aristoxenus, places these “last”:
Pythagoreans about 366/365 B.c.,** but the comic allusions would take:
us a few decades later; and there was a Pythagorean Lycon who wrote
after Aristotle.44 This Aristoxenus can ignore; such persons, he implies, |
are not really Pythagoreans.
All the same, it is not so easy to equate the Pythagoreans of Arise.
toxcnus with the mathematici and those of comedy with the acusmatiel,
According to Aristotle’s report, the mathematici recognize the Pythage
orcanism of the acusmatici, whereas Aristoxenus ignores conceptions
deviant from his own. The acusmata themselves do not enjoin complete
vegetarianism, but complicated rules of dietary detail; but in the
comedies absolute abstinence from meat seems to be prescribed. In
addition, there is an occasional mention of rhetorical, dialectical
writings, or speeches, as though they were regular Sophists.®
Neither Aristoxcnus nor Middle Comedy provides a wholly
undistorted reflection of a particular type of Pythagoreanism. The
former is contradicted by Eudoxus, who, as a personal pupil of Archy=
tas and himself one of the most brilliant Greek mathematicians, must
have been as close as anyone was to the tradition of the mathematic,
41.On the chronology: Epigraphical evidence shows that from 377 to 351, in order,
Antiphanes, Mnesimachus, Alexis, and Aristophon won victories at the Lenaca (IG
1{1Π2 2325), and Alexis won at the Dionysia of 347 ([G II/III? 2318, XI 278; sce A,
Pickard-Cambridge, The Dramatic Festivals of Athens [Oxford, 19687] 106, 114). The
years 408/40§-334/331 are given as the dates of Antiphanes. Alexis was long-lived, and
seems to have been active still about 270 μια (fr. 244). Antiphanes’ Neottis is to be dated
shortly after 342 (above, ch. II 3, n. 106), Aristophon’s Pythagoristes perhaps in 338 (Ed-
monds on fr. 11); Mnesimachus’ frr. 7-8 take us to the year 345. Antiphanes’ Coryeus
should perhaps be dated 359, Mnesimachus’ Aleneon ca. 340, Alexis’ Tarentint between
330 and 320 (see T. 1}. L. Webster, Studies in Later Greek Comedy [Manchester, 1953] $3f).
4 Aristoxenus’ Pythagoreans bathe, too (lam, VP g&).
43 Diod. 1§.76.4.
44 Below, 1. 64.
4 Alexis fro 2a: Uv@ayomopot καὶ λόγοι λεπτοὶ διεσμιλευμέναι τε φροντίδες...
Cratinus fr. 7: the Pythagorists drive ordinary people out of their minds with thelr
rhetorical subtletion, τοῖς ἀντιθέτοις, τοῖν πέρασι, τοῖν παρισώμασιν, τοῖς ἀποπλάνοιν,
τοῖς μεγέθεσιν νυυβυστικῶν.
5. Acusmatici and Mathematici
atid who states that Pythagoras abstained from ἔμψυχα and avoided
association with hunters and butchers.4® He recognizes, or perhaps
tuther exaggerates, the abstinences. Thus Aristoxenus appears to be
ihissociating himself from Eudoxus and the mathematici, insofar as he
ilenies the abstinences and does his best to present Pythagoras’ way of
life as normal.
‘The comedies, on the other hand, make everything cruder than it
was. At least one hint—not conclusive, to be sure—suggests that more
precise regulations were mentioned, in the matter of abstinence from
imeat.4?7 The main point here is, though, that comedy is not the place
for hairline distinctions. Its effect comes from mixing up the incom-
patible. It is tempting to conjecture that there were Pythagoreans, in
the first half of the fourth century, emigrating from southern Italy to
Athens,48 although there are certain difficulties in this.4° This much is
certain, though, that Plato, who had achieved extraordinary fame
inside and outside of Athens, and who was also often alluded to in the
comedics,®® traveled several times to Sicily and Magna Graecia, and
lad an especially close relationship with the Pythagoreans of Tarentum.
Also, his disciples equated their own philosophy with that of Pythag-
uras and in various books presented a new picture of the Samian sage.
‘Vo this extent, Pythagoras was a live topic of conversation in the
Athens of the fourth century, and it seems likely that the mockery of
the ‘ Pythagorists”’ was aimed at the Academy as well. More precisely,
when the comic poets identified the loftiest philosophical attempts of
the Platonists with the common, mendicant Pythagoreanism, they
were ridiculing one group as well as the other.*
The distinction between acusmatici and mathematici, drawn in the
ieports of the schism, is not directly reflected in the contradictions of
4 Hr, 36 Gisinger = 325 Lasserre = Por. VP 7.
4 Alexis fr. 199 = Gell. 4.11.8 (taken along with Aristox. fr. 25): “tac de animalibus
Alexix ctiam poeta in comoedia quae Πυθαγορίζουσα inscribitur, docet.” To judge by
the context, the eating of certain kinds of meat must have been mentioned as being
lis it.
Von Fritz, Pol. 76f, brings the appearance of the Pythagorists into connection with
the (ace that about 388 Dionysius I subjugated the southern Italian alliance of Croton,
‘ybaris, and Caulonia, and also with the report of Aristoxenus about the emigration
from Italy of all Pythagoreans except Archytas (fr. 18, lam. VP 2sof; the text is corrupt
aul cannot be restored with certainty; cf. von Fritz, Pol. 103f and, contra, Wehrli 53;
alae see Aristox. fr. 2, and von Fritz, Gnomon 32 [1960] 495).
‘The Pythagorists of Alexis and Cratinus come from Tarentum, but no Pythagorean
hd to leave the city of Archytas. Also, the comedies are some decades later than 388.
See Weiher 45-55.
"The plays called ‘Vareatini, of Alexis and Cratinus, are connected with Plato's Sicilian
journey by Edmonds, and dated about 360 (a different answer in Webster, above, n.
41) ‘The πανήγυρις comparison of Alexis Tarentini fr. 21g could be derived from) Hera-
clides fire, 87-88 (Burkert, Hermes ty6o, 165 1. 3).
Π. PYTHAGORAS IN THE EARLIEST TRADITION
the pictures of Pythagoreanism drawn by Aristoxenus and by the
comic poets; but it does seem to be clear that alongside of the scientific
Pythagoreanism with which Plato and his pupils allied themselves,
and which Aristoxenus stubbornly claimed to know best, there was
another quite different type of Pythagoreanism whose adherents were
mendicant “‘teetotalers’’ with special hopes about the next life.
The first evidence for this tendency is the Telauges of Aeschines of
Sphettus. Dittmar has elicited the fundamental facts about it from the
exiguous fragments :*? Telauges was depicted as a representative of the
Pythagorean life, characterized by a shabby poverty proudly displayed,
decked out with a κῴδιον and beggar’s scrip.
We can get a clear picture of Diodorus of Aspendus, as a historical
person of the first half of the fourth century B.c.53 A mocking verse by
the musician Stratonicus claims that with a “crazy garment of skins”
and “arrogant presumption (ὕβρις) he gathered followers.54 Arche-
stratus, in his “Ηδυπάθεια, recommends shark meat as a delicacy, and
says that anyone who will not eat the flesh of a creature that itself eats
human beings should restrict himself to vegetables, should seek out
Diodorus the Sage, and live the ascetic Pythagorean life with him.®
Here Diodorus is the well-known Pythagorean, famous for his vege-
tarianism. Timaeus mentions the striking clothing he wore, and
Sosicrates describes it more precisely; long hair, long beard, folded _
τρίβων, wallet and staff, “though the Pythagoreans before him wore
shining bright clothes, bathed and anointed themselves, and had thcir
hair cut according to the fashion.”** Athenaeus, without naming a
source, refers to him as barefoot and dirty.
This picture of Diodorus is the conventional picture of a Cynic, so
52 Dittmar, Aischines 213ff, von Fritz, RE V A 194ff, Pol. 76 n. 30.
53 The most important testimonia, from Stratonicus, Archestratus, Timaeus, and
Sosicrates, are in Ath. 4.163c-f; cf. Tannery MSc VII 201-210. Stratonicus died about
350 B.c. (Wilamowitz, Ind. Schol. Gott. 1893/1894, 16.1, Maas, REIV A 326f). Timaeus
cites him (FGrHist 566F16). Archestratus wrote about 330 B.c. Zeller (I 426 n. 3) confused
the citation of Archestratus with a nearby citation of Timon, and therefore mistakenly
dated Diodorus in the 3rd century. W. Crénert, RhM 62 (1907) 311f (dubiously Powell,
Coll. Alex. p. 212), suggested attributing the Stratonicus verses to Cercidas; but he could
scarcely have been cited by Timaeus, to say nothing of Archestratus.
54 Timaeus FGrHist §66F16 = Ath. 4.163¢-f: τῷ περὶ θηροπέπλου μανίας ὕβρεώς τε
περιστάσιμον στοὰν ἔχοντι Πυθαγόρου πελάτᾳ.
55 Fr, 23.18ff Brandt = Ath. 4.τ624--Ὸ:
wore πρέπει καθαρῶς ὁπόσοι τάδε μωρολογοῦσι
τοῖς λαχάνοις προσάγειν καὶ πρὸς Διόδωρον ἰόντας
τὸν σοφὸν ἐγκρατέως μετ᾽ ἐκείνου πυθαγουρίζειν.
56 Ath. 4.163f and 1}.1.. 6.14. This account of Sosicrates, who scems to use Aristoxenus
(ch. If 3, n. 249), is in turn the basis of Schol. ‘Vheocr. 14.5 (DK 1 478.376, above, n. 6),
on the differentiation between Πυθαγόρειοι and {Πυθαγορισταί, Long hair is also found
in the Pythagoras legend: ὁ ἐκ Ldyeou κομήτης, Tan. VP ott, with the parallel passages.
5. Acusmatici and Mathematici
that the usual thing is to dispose of him quickly as one who managed
“to represent his Cynicism as Pythagorean philosophy.’’*’ Or we may
read that “τῆς ascetic trend caused many Pythagoreans, after the rise
of Cynicism, to assume the Cynic garb and way of life.”®* Thus
Diodorus would be a Pythagorizing Cynic or a Cynicizing Pythag-
orean—in any case some kind of hybrid. But this ignores the chrono-
logy. The testimony of Stratonicus, who died about 350 B.c., shows
that we should regard Diodorus as in any case contemporary with,
but more probably, earlier than, Diogenes. According to the tradition,
Diodorus’ rival for the honor of having invented the Cynic costume
was not Diogenes but Antisthenes.®® If, as modern scholars rightly
believe,®° Antisthenes was not yet really a cynic, then Diodorus’ garb
was not Cynic either; rather, the Cynicism of Diogenes is in a way a
continuation of Pythagoreanism of the “acusmatic”’ stripe. There are
unmistakable coincidences—the praise of πόνος and contempt for
ἡδονή,51 and in general the choice of a special Bios in contrast with the
“normal” hit-or-miss life-style of the ordinary man. Just as the
Pythagorean feels himself a stranger on the earth, so the Cynic tries to
free himself from all ties. Threads lead from here to the Stoa as well;
both Zeno and Chrysippus were interested in aspects of Pythagorean-
ism 8
But Timaeus emphasizes that Diodorus “ pretended”’to have associ-
ated with Pythagoreans, and Sosicrates declares that the appearance of
Diodorus was a novelty and that before him the Pythagoreans had lived
ἃ more normal life. Finally, there is a statement in Iamblichus, probably
derived from Timaeus,® that after the catastrophe of the school
| iodorus of Aspendus was taken into the Pythagorean society “ because
of the need for members,” and that, returning to Greece, he published
the “Pythagorean sayings” —obviously meaning the acusmata.*4 The
tendency of these reports is obviously to deny real Pythagoreanism to
57 Zeller I 426 (where Diodorus is wrongly dated; see above, n. 43).
55 Ueberweg-Praechter 64. Cf. Rostagni, SerMin II 1.41.
ὃν D.L. 6.13 (Diocles and Neanthes name Antisthenes, Sosicrates names Diodorus of
Aspendus; above, n. 56).
“0 Cf, Wilamowitz, Platon II 163 ἢ. 4; J. Geffcken, Griechische Literaturgeschichte I
(Heidelberg, 1934) 30 and nn.; E. Schwartz, Charakterképfe aus der Antike® (Leipzig,
1943) 116ff, esp. 123f. On the history of Cynicism, see K. von Fritz, Quellenuntersuchungen
zit Leben und Philosophie des Diogenes von Sinope, Philologus Supp. 18.2 (1926); Ὁ. R.
Dudley, A History of Cynicism (London, 1937).
“! Above, ch. II 4, n. 14; for the comparison of ψυχή and ἁρμονία see D.L. 6.27, 65.
® Zeno wrote Πυθαγορικά (1).L. 7.4); Chrysippus cites a Pythagorean verse (Gell,
7.2.12 - SVFID294 Carm, aur. $4).
“3 Jain. VP 266; cf. Delatte, Musée Belge 1920, Rostagni, SaMint ΠῚ 1.41.
" διέδωκε τὰς Π|υθαγωρείους φωνάς; for φωνή, “saying,” see Pl Prot, aqib: τὴν
Σιμωνίδου dari... , Epicurus Ep. 1.10.
IT. PYTHAGORAS IN THE EARLIEST TRADITION
Diodorus, though Iamblichus admits that he had been a member of
the school. But the two oldest witnesses, Stratonicus and Archestratus,
call him a Pythagorean, without qualification. Sosicrates, however, is
dependent on Aristoxenus (above, n. 56), and Timaeus is in the same
line. Thus what first was generally regarded as Pythagorean in the
fourth century is later branded as “alleged” Pythagoreanism, and
contrasted with that which is “genuine.”
With Diodorus doubtless belongs Lycon,® who criticized Aristotle’s
extravagant way of living. His ideal must have been something in the
nature of Cynic self-sufficiency. Aristocles introduces him as one “ who
called himself a Pythagorean,” showing the same sort of reserve about
his Pythagoreanism as Timaeus did about that of Diodorus. There is
no reason not to identify him with the Lycon of Iasus who wrote on
the Pythagorean life, emphasizing the “moderate” regimen of
Pythagoras.** After Lycon, who must have been approximately con-
temporary with Aristotle, there are no more Pythagoreans of this type
to be found. An echo of such activity can still be heard in Onesicritus,
who named Pythagoras as one of the Greeks who, before Socrates and
Diogenes, had taught doctrines like those of the Indian Gymno-
sophists.6’ But Onesicritus is regarded as a Cynic. The tendency in
Pythagoreanism represented by Diodorus of Aspendus was absorbed
by Cynicism, which took shape as the form of the self-sufficient,
world-despising Bios which suited the demands of the age. Meanwhile
the spiritual power of Pythagoreanism found, through the interpreta-
tion of the Platonists, a new vehicle adaptable to changing times.
In the fourth century, then, alongside those Pythagoreans with
65 Aristocles ap. Euseb. Praep. evang. 15.2.8 (DK $7.4; I. Diiring, Aristotle in the Ancient
Biographical Tradition [G6teborg, 1957] Test. 58 1; cf. p. 391). Zeller (I 426 n. 3) connected
Lycon with Diodorus.
66 Ath. 10.418e (DK $7.3): Δύκων ὁ ᾿Ιασεὺς ἐν τῷ περὶ Πυθαγορείου βίου. Capelle doubts
the identification (RE XII 2308f, s.v. Lycon n. 15). Iam. VP 267 (the catalogue of Pythag-
oreans) lists a Lycon as Tarentine, D.L. 5.69 knows only one Πυθαγορικὸς Λύκων.
This Lycon is naturally to be distinguished from the historian Lycus of Rhegium (earlier
than Timaeus: FGrHist 570, and Jacoby in IIIb Komm. $97ff; also Laqueur, RE s.v.
Lykos, no. §0)—who, considering his interest in miracles, asceticism, and the like (frr.
5, 6, 7), certainly must have mentioned Pythagoras.—Por. VP 5 cites Aedxos ἐν τῇ τετάρτῃ
τῶν ἱστοριῶν. The emendation Avxos is tempting, but the second person σοι is not very
likely in a work of history. Jacoby classes the passage among the doubtful fragments
of Lycus (FGrHist §60F15) and is inclined to assign it to Lycon (IIIb Komm. 601). The
ἱστορίαι cannot be the same as the περὶ Πυθαγορείου βίου, since it only takes up the origin
of Pythagoras in the fourth book, There can be no certainty about the attribution in the
case of the botanical data in IK $7.5 (Lycon) and 57.2 (where the MSS have [bycus;
there were also physicians named Lycus, RE s.v., nos. $1 and §2).
8 EG rblist 1947 --- Strabo 1§.716 (Dittmar saw the relevance of this to the “ Pythag-
orist” contest, Aischines 217). ἐμψύχων ἀπέχεσθαι. appears among the rules of
Afoka: Journ. Asiat. 246 (1998) UE
5. Acusmatici and Mathematici
whom the Academy felt akin in their philosophical and scientific
endeavors, there were others of a quite different type, exemplified by
Diodorus of Aspendus but presupposed also by Aeschines and the comic
poets. Their characteristic mark is not μαθήματα but a Bios. They
remind us of the acusmata (alluded to in the tradition: above, n. 64) by
their avoidance of baths and of shoes, and by their vegetarianism,
though the evidence, almost entirely satirical and negative, oversimpli-
fics and distorts the picture. Allusions in comedy permit us to attribute
to them a belief in metempsychosis and hopes for a better life to come.
Aristoxenus acts as though this kind of Pythagorean did not exist,
though his portrait of Pythagoras is specifically intended to correct the
impressions they made. From Timaeus on, these are “‘alleged”’ Pythag-
orcans, distinguished from “genuine” ones; and from Timaeus on the
tradition becomes canonical that the real teaching of Pythagoras was
esoteric, as distinguished from the imperfect, exoteric preliminary stage.
Aristoxenus is corrected by Eudoxus in one point, and in a way this
corroborates Iamblichus’ account of the division. This report reveals
details that were obscured from the time of Aristoxenus and Timaceus,
for tendentious reasons. This confirms its Aristotelian origin, and it
must be taken seriously as an expression of historical facts. The
iathematici, whose successors Plato and his disciples thought of them-
sclves as being, tried to regain the ancient wisdom of Pythagoras by
scientific studies of their own, while acusmatici like Diodorus of
Aspendus wished only to live a straitlaced life in accordance with the
ancient precepts. The mathematici, followed in this by Eudoxus, did not
attack the ritual observances taught by Pythagoras, but the acusmatici
saw a defection from Pythagoras in the further development of
scientific study. But neither tendency could endure except in altered
form: “mathematical” Pythagoreanism in the reinterpretation of the
Platonists, and the “acusmatic’’ way of life, rationalized and secularized,
in Cynicism.®® And, since the “mathematical” tradition, in its Platonic
metamorphosis, became completely dominant in the literary realm,
the contention of the mathematici also won out, that the acusmatici were
not genuine, but only imperfect, Pythagoreans.
68 On the question of the continuity of the Pythagorean tradition, see Burkert, Philologus
1961. In the Hellenistic period, or, say, in the 3rd and 2nd centuries B.c., there do not
scem to have been people calling themselves Πυθαγόρειοι, nor is there evidence for the
survival of Pythagorean cult, ὄργια of the acusmatici, which are commonly assumed to
have continued. It does seem, however, that there was a good deal of interest in Pythag-
oreanisni: and this was part of the reason for the apocryphal literature. The [Tu@ayopixras
from Athens in Theocritus (14.5), barefoot, pale, and hungry, may be a figure from
comedy (Wilamowitz, Platon IT 84).
II, PYTHAGORAS IN THE EARLIEST TRADITION
Hippasus,*? who plays a crucial role in the versions of both groups,
is mentioned in certain early reports. Aristotle and Theophrastus
attribute to him, as well as to Heraclitus, the doctrine that fire is the
ἀρχή. Ancient scholars drew a chronological conclusion from the fact
that his name comes first, but this is hardly justified.”° Aristoxenus
credits him with an acoustic experiment and brackets him with Glaucus
of Rhegium.” Iamblichus names him three times, along with Archytas,
as discoverer of the “harmonic mean,” again a connection with music
theory.” According to the report of the schism, Hippasus claimed to
have discovered the dodecahedron and was drowned at sea.7? He
wrote no books.’4 The evidence seems to point toward the first half of
the fifth century—including an apocryphal report that he was the
teacher of Empedocles.”> His home city was Metapontum,” and there
is a slight trace of a connection with Phlius.”?
Thus Hippasus is the oldest Pythagorean we know of who worked
at mathematics and music theory, and also had something to say in the
realm of natural philosophy—though, to be sure, not in terms of a
theory of number, or of a philosophy of “limit,” “unlimited,” and
“harmony.” The independent reports of Aristotle and Aristoxenus
confirm the account of the division into sects—in the version of De
communi mathematica scientia—according to which Hippasus was one of
°° DK 18. Cf. Zeller I 603f, Wellmann, RE VIII 1687f (too brief), Frank 261ff (hyper- _
critical), Timpanaro Cardini 78ff (harmonizing).
7° Arist. Met. 98427, Theophr, Phys. op. fr. 1 = Dox. 475; other doxographical data
DK 18.7-9. On Hippasus as teacher of Heraclitus, Suda s.v. Heraclitus (DK 18.1a). On
the soul as fire, DK 18.9; Arist. De an. 416a9.
*? Aristox. fr. 90 = Schol. Pl. Phd. 108d; on the facts, see below, ch. IV 1. In the
comment on the source, μέμνηται δὲ τούτων ’Aptardééevos..., DK follows Hermann
in printing τούτου instead of the MS reading τούτων (Greene, Scholia Platonica; von
Pritz, AnnMath 1945, 245; Wehrli), which alone gives the testimony its importance:
Aristoxenus mentioned not only Glaucus but the whole incident, including Hippasus.
It is uncertain how Aristoxenus dealt with the contrast between Hippasus’ music theory
and his own (cf. Wehrli, Aristoxenos p. 77).
72 DK 18.15; below, ch. VI 2.
Tam. VP 88, with Comm. math. sc. 77.18ff; cf. lam. VP 247; below, ch. VI 3. The
death of the offender at sea reminds of the death of Protagoras (on which see von Fritz,
RE XXIII orof).
** Demetrius of Magnesia ap. D.L. 8.84 (DK 18.1); cf. below, n. 78.
Ὁ Τῆς “letter of Telauges to Philolaus,” cited by Neanthes (FGrHist 84F26 == D.L.
8.55, Thesleff, Texts p. 189). Even Neanthes doubts the authenticity of the letter, but in
any case it was in existence by about 200 n.c.
8 Arist. Met. 98427, D.L. 8.84, etc. He is from ‘Croton or Metapontum” according
to Tam. VP 81 = Comm. math. sc. p. 76.23; from Sybaris according to the catalogue Iam,
VP 267. (This would date him after 444).
τ There are two versions of a genealogy of Pythagoras which connect him with Philius:
Cleonymus of Phlius-Euthyphro-Hippasus Marmacus Pythagoras, according τὸ DLL.
8.1 Hippasus of Phlius Euphron Mnesarchas Pythagoras according. τὸ Paus. 2.13.2.
These genealogies obviously stem from the Pythagorean group in Philius. That the
UncOmmEn name Hippastis appears in both ix hardly an accident.
5. Acusmatici and Mathematici
the mathematici. This report shows the position of Hippasus between
the fronts which were later formed. For the acusmatici his activity was
something new, and subversive; but the mathematici, in order to find a
firm basis for new doctrines, were forced to abandon Hippasus and
brand him a plagiarist. Since he became in this way a scapegoat for
both sides, it was easy for further charges to emerge, such as that he
had traduced Pythagoras.7®
It is impossible to determine whether the breach between Hippasus
and the other Pythagoreans came before or after the political catas-
trophe about 450 B.C.; it is tempting to suppose that there was a
connection between the inner and outer crises of Pythagoreanism.7®
But it is a more important question, which of the two opposed con-
ceptions of Pythagoras should be considered correct. The name
μαθηματικοί, interesting as it is, scarcely helps;8° we must weigh the
18. According to Apollonius (lam. VP 257ff) Hippasus was one of those who rebelled
against Pythagorean dominance. Ninon produced a λόγος ἱερός, πεπλακὼς Kai γεγραφὼς
ἐξ ὧν μάλιστα αὐτοὺς ἤμελλε διαβάλλειν (lam. VP 258); and there must be some connec-
tion between this and the remark of Heraclides Lembus, in his enumeration of Pythagorean
writings τὸν δὲ Μυστικὸν λόγον ᾿Ιππάσου... εἶναι, γεγραμμένον ἐπὶ διαβολῇ Πυθαγόρου.
Either Heraclides is dependent on the story which later appears in Apollonius, only that
Hippasus, the apostate, is substituted for Ninon; or the narrative is late and there was an
actual ἱερὸς λόγος which was regarded as overly primitive and therefore discreditable,
so that it was branded a forgery and a libel and put down to the account of the wicked
Hippasus. In this case, however, Hippasus would not be an acusmaticus (as Timpanaro
Cardini 81), but a mathematicus who wished to conceal his plagiarism and characterize
I’ ythagoras as a primitive—as scandalous letters of Epicurus were forged not by Epicureans
but by a Stoic (D.L. 10.3). Cf. Frank 70.
7° Tannery (Géom. 85f), relying on the account of Apollonius in Jam. VP 257ff, thinks
that the internal division of the school started by Hippasus finally led to civil war and to
the final catastrophe. Tannery speculates (MSc VII 209f, on Diodorus of Aspendus) that
after the collapse of their political strength the Pythagoreans restricted themselves to
their religious and superstitious activities; against this is the fact that the mathematici
whom Plato knew were all active after the political catastrophe.
80 Whether the word μαθηματικός “concerned with the subjects of learning,” came
to have its narrower sense “mathematician” precisely among the Pythagoreans (von
Fritz, SBM 1960, 20f) or only in the Academy (below, ch. VI 1), the history of the word
cannot decide whether the mathematici were descended from Pythagoras or Hippasus.
Perhaps the terms μαθηματικοί and ἀκουσματικοί do not go back to the original schism,
but were only later applied to the rival groups. Formations in -ἰκός become frequent from
Plato on. (A. Amman, -ἰκός bei Platon, Diss. Fribourg, 1953; the earliest occurrence of
μαθηματικός is Pl. Soph. 219c). The primary contrast is between ἀκούσματα and μαθήματα.
It was in the educational movement of the sth century that the word μανθάνειν acquired
its specialized sense. Parmenides (frr. 1.31, 8.52) and Empedocles (fr. 17.14) call for μανθάνειν,
and Protagoras wrote Περὶ τῶν μαθημάτων (D.L. 9.55; the sense is “On the Branches of
learning,” and it is a mistake to apply it to mathematics, as in DK 80B7; cf. Pl. Prot.
318d and 313¢, Laches 17y9¢e, 182d). In the Clouds the μαθητής is an established type. In
Pythagoreanism, “hearing” plays a prominent part, and this too suggests that ἄκουσμα
is the older idea (above, ch. ΠῚ 4, 1.76). Uf, as tine went on, certain Pythagoreans empha-
sized the importance of μαθήματα as contrasted with ἀκούσματα, they were integrating
themselves into the intellectual development of the sth century: they do not want
“hearsay” but the comprehension of truth,
II. PYTHAGORAS IN THE EARLIEST TRADITION
content of the two versions. The mathematici claimed that their studies
were no more than explication of the doctrine of Pythagoras, and that
“everything came from him.” This principle, which seems to have
been accepted in the day of Kenocrates, emanates from a completely
unhistorical point of view, close to that of myth. What seems impor-
tant and desirable, takes the form of a quasi-historical assertion: in the
beginning, there was Pythagoras.
On the side of the acusmatici is the fact that the acusmata, the ideas of
metempsychosis, and the legend of Pythagoras are early attested and
primitive in character. The mathematici, in their effort to get free of
this annoyance, resorted to the suspicious expedient of secrecy and
esoterism. |
If, in spite of this, efforts continue to be made to show that the
acusmatici were not entirely in the right, that Pythagoras was “not
merely” a kind of shaman, and that science too was present in
Pythagoreanism from the beginning, at least in embryonic form, at
the base of these efforts lies not only the force of the dominant tradi-
tion, but also the seemingly ineluctable idea of the law of development,
the postulate that the later Pythagoreanism must in some way be
foreshadowed in the earlier. Yet one may well ask just how much that
is coherent and stable is necessary to explain the development. The
question of what is scientific depends more upon form, method, and .
proof than upon the content or the practical function. May it not be
that the conceptual and scientific impulse simply provides a new form
for an ancient and pre-scientific lore or attitude? The “wisdom” of
the acusmata, the “wisdom” of a shaman-like “divine man”’ can stand
without the prop of science, and did so in the activity of the acusmatiel
down into the fourth century. Greek science, including Greek mathe-
matics, may well have another and non-Pythagorean origin.
6. EARLY EVIDENCE FOR PYTHAGORAS AS A SCIENTIST?
In discussing Pythagoras scholars have always, with varying degrees
of uneasiness, taken cognizance of his doctrine of metempsychosis and
the accounts of his miracles.! But the principal endeavor of the historians
of philosophy, in this area, has been to show that this aspect of the man
is not the only one, and not even the most important one; they have
1 ‘ ᾿ , .
Jaeger asks (Paidelal 221-1162 Eng. ed.), “ But what connexion has all this [number
theory, geometry, music theory, and astronomy] with che doctrine of transmigration. ..?"
6. Early Evidence for Pythagoras as a Scientist?
looked to find at least the “germ” of the mathematics, natural science,
and philosophy that were current, in later times, under the name of
Pythagoras. Often a simple “not only-but also” has seemed enough;
he was not only a “medicine man” but also a thinker. But may not
even a “shaman” perhaps accomplish intellectual feats, without
necessarily clothing them in strictly rational or conceptual form?
More penetrating are the efforts to discern an inner connection between
apparently heterogeneous things.
The point of departure for these must always be the little group of
testimonia which, in praise or in irony, speak of the preeminent,
comprehensive knowledge of Pythagoras. First comes Heraclitus:
πολυμαθίη νόον ἔχειν οὐ διδάσκει. “Ησίοδον yap ἂν ἐδίδαξε καὶ [Πυθαγόρην
αὖτίς τε Ξενοφάνεά τε καὶ “Ἑκαταῖον (fr. 40). Πυθαγόρης Μνησάρχου
ἱστορίην ἤσκησεν ἀνθρώπων μάλιστα πάντων καὶ ἐκλεξάμενος ταύτας
τὰς συγγραφὰς ἐποιήσατο ἑαυτοῦ σοφίην, πολυμαθίην, κακοτεχνίην
(fr. 129). Then Empedocles (fr. 129):
> ἃ , > ,
.. ἀνὴρ περιώσια εἰδώς,
43 4 7 ᾽ ? 4 “-
ὃς δὴ μήκιστον πραπίδων ἐκτήσατο πλοῦτον,
παντοίων τε μάλιστα σοφῶν «τ᾽» ἐπιήρανος ἔργων...
And Ion of Chios:?
μι / > 4 4 54] ‘ i
εἴπερ Πυθαγόρης ἐτύμως σοφός, ὃς περι πάντων
ἀνθρώπων γνώμας εἶδε καὶ ἐξέμαθεν.
ΕἸ]οτοάοτζας tells of Zalmoxis, who was cleverer than the Thracians, οἷα
“Ελλησί τε ὁμιλήσαντα καὶ “Ελλήνων οὐ τῷ ἀσθενεστάτῳ σοφιστῇ Πυθαγόρῃ
(4.95). The key terms are, then, πολυμαθίη, ἱστορίη, and σοφιστής.
From the accusation of polymathy, “we may infer that the later
‘so-called Pythagoreans’ ... were right in naming Pythagoras as the
founder of their peculiar science.”? More must be involved than reli-
vious revelation, for the name of Pythagoras stands beside that of
Xenophanes. Surely, however, the idea of polymathy must apply to
all four of the men named in the fragment, and especially Hesiod, who
2 Cf. ch. II 3, ἡ. 13.
8 Jaeger, Paideia 1 221 -- I 162 Eng. ed. Rohde, too (Q τος; Psyche Il 19 = 374 Eng.
cd.), finds in this word the seed of later science. Cf. Mondolfo in ZM 316f, G. Vlastos,
Philos Q 2 (1042) 111. 64, K. Freeman, The Pre-Socratic Philosophers (Oxford, 19497)
76 (with this word Pythagoras was classified among, scientists by Heraclitus).—Against
these attempts to extract “science” from πολυμαθίη, see Reinhardt, Parm, 232ff, Frank
356 n. 166, W. J. Verdenius, Maenosyne 3.13 (1947) 280ff.
Π. PYTHAGORAS IN THE EARLIEST TRADITION
is the first mentioned; and in fact Hesiod’s Theogony and the Catalogues
are polymathy in the truest sense of the word. But Hecataeus and
Xenophanes are distinctly separated from the other pair by the word
αὖτις, suggesting that Pythagoras belongs more closely with Hesiod
than with them. This can scarcely be accounted for as a chronological
indication;* the difference in time between Hesiod and Pythagoras is
greater than that between Pythagoras and Xenophanes or Hecataeus,
Something in the nature of the subject matter, then, must be what
connects Pythagoras with Hesiod. Hecataeus and Xenophanes have in
common their emphatically modern and polemical attitude, critical
of the traditional mythology; Pythagoras does not belong in their
company. His “ much knowledge”’ spans the entire world of man and,
above all, the world of the supernatural, both this life and the next,
This is clear from Empedocles. He stands beside Hesiod as the repre-
sentative of Orpheus, so to speak—the representative of non-Hesiodie
mythical teaching about the world and its gods.
‘Ioropin is rightly regarded as a key word for the open-minded kind
of inquiry based on observation, pursued by the Ionians; so some have
seen Heraclitus as testifying to Pythagoras’ “‘scientific research.’ If it
only were not for those awkward συγγραφαί which sound very much
like “Orphic” writings, and the word κακοτεχνίη, which means
“ skulduggery’’!§ When one considers the formulation by Hermesi- - :
anax of Colophon (third century B.c.), “Ἡσίοδον πάσης ἤρανον ἱστορίης
(fr. 2.22 D.), one realizes the insubstantiality of all the inferences drawn
from the passage of Heraclitus. ‘oropin does not necessarily imply
anything more than the Hesiodic type of πολυμαθίη, and does not make
any definite allusion to rationale Wissenschaft, “science.”
The word σοφιστής is translated “scientist,” too;’ and it has even
been asserted that the word proves that Herodotus “knows nothing of
Pythagoras as a miracle-worker.’’® Here again the perspective is
distorted. In the fifth century it is poets, more than anyone else, that
* This interpretation is attempted by Lévy, Sources 2 n. 8; H. Thesleff, On Dating
Xenophanes (Helsinki: Soc. Scient. Fenn., Comm. Hum. Litt. 23.3, 1957) 7 (dating
Xenophanes late). H. Gomperz, Hermes 1923, 36 n. 1, and M. Marcovich, Philologus
108 (1964) 40, speculate that Pythagoras, unlike Xenophanes, was already dead, and
therefore classed with Hesiod.
ὅ Burnet, EGP ὃς and 97: “‘scientific investigation”; Rostagni, Verbo 131: “ricerca
scientifica’”’; Raven in KR 228: “scientific inquiry.”-—-On Comm. math. se. p. 78. =
Iam. VP 80, sce below, ch. VI τ.
8 Above, ch. ID 3, nn. 64, 228.
* Burnet, HGP &§: “scientific man,” 97: does imply scientific ideas.” Contra (rightly),
Rathmann 47, Viastos, Philos Q 2 (τ 42} 11 τ᾿. 64,
*Corssen, RAM 1yr2, 46.
᾿
6. Early Evidence for Pythagoras as a Scientist?
are designated σοφισταί," and even for Isocrates Homer is ὁ μεγίστην ἐπὶ
σοφίᾳ δόξαν εἰληφώς (Soph. 2). For Herodotus, along with Pythagoras,
successors of Melampus and Solon are σοφισταί. Melampus, who
according to Herodotus introduced the worship of Dionysus from
livypt, is without a doubt a Wundermann par excellence; and even if
one supposed that Pythagoras was closer to Solon than to him, he is
still far from science. The new thing that was introduced by those
whom Plato called Sophists is not at all suggested by the word.
“Wisdom” is an advantage admired and sought from time im-
memorial, but there are utterly different conceptions, from time to
time, as to what it consists of, what it pertains to, and how it manifests
itself. Were it certain that Pythagoras was a scientist, these words
could be understood in this sense. As long, however, as only his
“shamanistic” activities are early attested, our conclusion must be
that no statement about the “wisdom” of Pythagoras carries us
further than these activities. Quite the contrary, in fact: the Hippo-
cratic book On the Sacred Disease attacks the itinerant “medicine
men,” who expect to heal the sick with magic rites, μάγοι καὶ ἀγύρται
καὶ καθαρταὶ καὶ ἀλαζόνες (VI 354 1,1, and adds, προσποιέονται εὐ πλέον τι
εἰδέναι. The claim to know “more,” beyond the limits set for ordinary
men, is the mark of the shaman; τεκμαίρεσθαι is that of the scientist
(Alcmaeon fr. 1).
The efforts to discover a unity of scientific and religious thinking in
Pythagoras, and thus a step beyond the merely ritual and mythical
realm, center about the ideas of κάθαρσις and ἀνάμνησις. In both cases
the question of Plato’s position is difficult; that is, did he merely take
over these ideas, or reinterpret them in an independent manner? Since
the time of Déring it has been thought that the concept of κάθαρσις
holds the key to understanding the connection of religion and science
in carly Pythagoreanism." Rigorous scientific work, especially mathe-
inatics, is supposed to release the soul from its close tie to the body and
in this way becomes the principal agent of “ purification,” upon which
9 Pi, Isthm. §.36, with schol., Soph. fr. 820 N., Hsch. s.v. σοφιστήν, G. B. Kerferd,
ΟἿ 64 (1950) 8-10. This removes the necessity of interpreting the special position of the
‘wise man” in the metempsychosis doctrines of Pindar (fr. 133 Schr.) and Empedocles
(fr. 146) as related to the “scientific” character of Pythagoreanism (Mondolfo in ΖΜ
y24tf).
In Hde. 1.29, 2.49. In Eur. (Ὁ) Rhes. 949 Orpheus is σοφιστής, and Menander seems to
have dubbed Epimenides συφώτατος (Κωνειαζόμεναι fr. 2 Koerte - fr. 308 Kock).
"DGring, AGP 1892, $05, and, following him, Burnet (ΠΡ 97f), through whom,
especially, the idea became current. CE, for example, Delatte, Pol. ς, Rougier 1orff,
Robin, Pensée 65, Mondolfo in 7M 646, Corntord, PrSap trot.
II. PYTHAGORAS IN THE EARLIEST TRADITION
depends the well-being of the soul in this life and the next. This is
clearly stated in the Phaedo, which introduces Pythagoreans as interes
locutors and has a good deal to say about Pythagorean matters.t®
Separation of soul and body, “dying,” is the goal of all true philosophy,
so that pure truth may be apprehended by the soul, in its pure state
(64a)—xdBapors through φιλοσοφία (67cd). And in the Republic it is the
sciences of arithmetic, geometry, astronomy, and music that lead from
the world of appearances to the ideas and effect the necessary περιαγωγή
of the soul (521c). But here we must move carefully; this line of
thought is so closely bound up with the theory of ideas that without
this, it becomes meaningless. Without Plato’s χωρισμός, without the
view that there are incorporeal, nonsensible objects of knowledge and
science, all science must remain tied to the visible world, and this
commitment becomes deeper and deeper. And Aristotle attests that
the Pythagoreans were in this very situation.!® Among these Pythag-
oreans there is not a trace of the idea of science as an escape from the
world; it may not be a coincidence that at the point in the argument
of the Phaedo where Socrates first speaks of dying as the goal of life
Plato has Simmias laugh (64a). |
In fact, the idea of purification through science was apparently not
ascribed to Pythagoras before the time of Iamblichus.14 To be sure, an
important and widely cited sentence of Aristoxenus says of Πυθαγορικοὶ -
. καθάρσει ἐχρῶντο τοῦ μὲν σώματος διὰ τῆς ἰατρικῆς, τῆς δὲ ψυχῆς
διὰ τῆς μουσικῆς, and a number of similar reports may be found in
Iamblichus and others,'® but the topic here is music in the proper sense
of the word, which is understood in the light of magic and is bound
about by ritual. Science is neither involved in the origin of these usages
nor a necessary consequence of them—though by means of a new
interpretation of music it may naturalize itself in this realm.* There
is, however, no testimony to the occurrence of such a transposition
12 On Simmias, Cebes, and Echecrates, above, ch. I 4, nn. 39-40; on the doctrine of
ψυχή-ἁρμονία, below, ch. III 2.
13 Met. 989b29; cf. ch. I 2, n. 15.
Ἷ : :
* Theo Sm. 14ff is only concerned with Plato. The Lysis letter (Hercher, Epistologr, gr.
P. 602.12ff = Iam. VP p. 44.2ff, closely dependent on Pl. Rep. 429d) uses katharsis in a
ifferent sense, namely moral preparation for admission to the esoteric teaching.—In
detail, Iam. Comm. math. sc. 22, esp. p. 69.6ff, pp. $5ff, p.84.12
18 Ari ci thes : Oe
sees fr. 26; below, ch. Vir. The sharp distinction of soul from body re-
ΠΑΡΑ ον ea Beau later sources represent music as also acting on the body
am, ττοῦ, etc.), which doubtless corresponds more closely to tl iginal magic:
ene scly to the original magical
' Socrates in PL Phd. Gta: dy φιλοσοφίαν μὲν οὔσης μεγίστης μουσικῆς.
6. Early Evidence for Pythagoras as a Scientist?
before Plato. Festugiére derived the idea of purification, in Plato, from
yencral-Greck cult practices;"’ Boyancé replied with the significant
comment that the question was not about a single, ephemeral ritual
act, but about a lasting style of life, and that in this respect the
Pythagoreans seem to have been Plato’s predecessors."* But scientific
activity is not necessarily comprised in the Πυθαγόρειος τρόπος τοῦ βίου,
cither.
The situation is scarcely different with recollection (ἀνάμνησις) and
related ideas. Here, too, there is a discernible Pythagorean background:
Empedocles (fr. 129) says that Pythagoras, when he put all his spiritual
power to work, could survey ten and twenty generations. Surely this
assumes that he recalled his earlier incarnations. The later tradition
tells of a system of memory training among the Pythagoreans. They
tried in the morning, or in the evening, to recall all the events of the
past day, and even of the day before.!* In the world of the dead are
the springs of Lethe and of Mnemosyne, and the initiate 1s warned of
the former and directed toward the latter.?° So the attractive conjecture
has been made that the original goal of Pythagorean exercises in con-
centration was to enhance the strength of the soul so that, following
the example of Pythagoras, it may avoid the spring of Lethe.?! Perhaps
it was also customary practice in the mysteries to awaken in the initiate
a ‘‘memory” of his divine descent.
The concept of reminiscence is basic to Plato's epistemology. All
truc knowledge, as knowledge of an idea, is a process of reminding
oneself of something that the soul has seen before its entry into the
body. To make this clear, Plato makes use of the doctrines of metem-
psychosis, and he demonstrates the existence of anamnesis, with the
help of mathematics, in the famous passage in the Meno (80dff).
Because of the tendency to regard everything mathematical as eo ipso
‘7 A. J. Festugiére, Contemplation et vie contemplative selon Platon (Paris, 1950°) 123ff.
1® REG 1941, 164 n. 3; above, ch. If 4.
liod. 10.5.1, Iam. VP 165 (in the morning), Cic. Sen. 38. The verses cited at Por.
ΕΓ 40 are the same as Carm. aur. 40ff; one also cited at D.L. 8.22 (evening). The source
it this tradition cannot be determined with certainty (on Delatte’s reconstructed ‘Tepds
λόγος see below, ch. III 1). See also Apollonius ap. lam. VP 256. According to Th. ar.
41.17, the number τὸ is Μνήμη and 1 is Μνημοσύνη; below, ch. Il 2, n. 48; above, ch. II
A, 24.
ἡ (Cf, above, ch. I] 3, nm. 189.
4 Piogenes Antonius ap. Por. VP 45: ὡς ἀθάνατος ἡ ψυχὴ καὶ τοῖς κεκαθαρμένοις
εἰς μνήμην τοῦ παλαιοῦ βίου ἀφικνεῖται; Proch. In. ‘Vim. 1124.5; cf. Hynm. Orph. 77.9:
eis Μνημοσύνην: μύσταις μνήμην ἐπέγειρε εὐιέρου τελετῆς. Also Emp. fr. 119; Dieterich,
Nekyia 122; Rohde, Psyche 186 πὶ Ρ. 406 0. 96 Eng. ed.; Dodds, Irr. 173 ἡ. 107;
1 2, Vernant, “Aspects mythiques de la mémoire en Gréce,” Journal de psychologic
(iggy) 1; B. Gladigow, Hermes y§ (1y67) 407-420.
IT, PYTHAGORAS IN THE EARLIEST TRADITION
Pythagorean, all of this is commonly thought of as Pythagorean,” and
if true this would show a basic correlation of metempsychosis, genuinely
philosophical epistemology, and exact mathematics.
But a closer look reveals that the connection of Pythagoras with
Plato, in relation to anamnesis, is scarcely more than an equivocation,
In Plato, what is “discovered” is that group of elements of human
knowledge, beyond the empirical and not discoverable in experience |
which are called in modern philosophy the a priori. They can be om ,
most clearly in the realm of mathematics or in expressions of relationship
like “equality.”"** Therefore, in Plato mathematics is a means of proof,
and reminiscence as well as metempsychosis is a myth-like clarification
of facts of the greatest philosophical relevance. Plato makes use of
mathematics because it is necessary to his doctrine, not because of any
partiality toward Pythagoreans or mystical attraction toward number,
The intended goal of his argument is the theory of ideas, for it is in the
ideas that Plato finds that knowledge, beyond the empirical. The
object is not the extension of human experience to “ten and twenty
human lifetimes,” nor the projection of this life into another, very real
next life, but a vision of a quite different kind. The knowledge of the
ideas is supramundane, and cannot be attained by living through even
a large number of earthly lives. This is made clear in the myth of the
Phaedrus, and the Meno hints at it. Any slave can “recall,” because
what is at issue is not just any experience, but that which is generally
valid; it is explicitly stated that this slave must have attained thig
knowledge “when he was not a human being” (Meno 86a). Here we
are in a realm which, according to Aristotle, was foreign to the
Pythagoreans: διαλέγονται μέντοι καὶ πραγματεύονται περὶ φύσεως πάντα
(Met. ο89033). It is impossible to credit Pythagoras’ eschatological
22 Burnet, ThPI 43: “I see no difficulty in referring this doctrine in its mathematical
application to Pythagoras himself.” He is more reserved in the edition of the Phaedo
(72), since, as represented by Plato, Simmias and Cebes do not seem familiar with the
doctrine; therefore the doctrine is attributed to Socrates: ‘he applied the old religioun
doctrine of ἀνάμνησις to science.” Here the difference is seen; we only need to replace
Socrates” with “Plato.”’—The matter is most fully discussed by Cameron ae han
the twofold thesis that ἀνάμνησις in Pythagoras comprehended ‘a theory of kriowled ο"
and that “divine knowledge was Number” (20). Neither of these points. is Seuabi
(cf. Cherniss, AJP 61 [1040 361), and in particular there is no evidence that forche Ρ tha -
oreans “recollection” was bound up with number, (Heraclides fr. 44. is surely thinkin
along Platonic lines; cf ch. ΠῚ, αν. 78.) Long (08) emphasizes that the doctrine at
transmigration as taught in the Meno is dependent on Pindar, but finds, in the nearh
mathematical discussion “additional proof" that Pindar, too, had α P tha rorcan ba ᾿-
ground; mathematical, ergo Pythagorean, (On this see ch, VI) pvt ates
| τ So PI. Mid, 72; brief treatment of recollection Phdr. δον ον differently Rep. si8b-d
ΠΗ. "480. τ: Bo 1 ἢ, Hermes ot (1064) γχ0. $2. Phoclus repards the doctrine πᾶσα
μάθησιν ἀνίμνησιν as Pythaporean (ln Back p. 4.9).
6. Early Evidence for Pythagoras as a Scientist?
doctrine with Plato’s mathematics and his theory of ideas. The
development of the concept of anamnesis nto epistemology and the
proof of it by mathematical means are seen to be typical “Platonic
transposition.”’24 What is Pythagorean 15 practice in concentration and
memory, which have no necessary connection with mathematical
science, but belong to the world of © shamanism.”
Generally speaking, the argument from silence is not very persuasive.
And yet one is impressed by the negative consensus of the ancient
sources, down to Aristotle, in the matter of Pythagoras the philosopher
and scientist. Herodotus mentions him in connection with Zalmoxis,
and mentions Pythagoreanism in speaking of the rituals of Egypt; but,
when he deals with the rise of geometry in Egypt (2.109), Pythagoras’
name does not occur. Democritus was acquainted with Pythagoreans,”
and wrote a book entitled Πυθαγόρης, but it is classified, in the cata-
logue of Democritus’ writings, among the ethical tetralogies, along
with Περὶ τῶν ἐν “Αιδου.38 The Dissoi logoi, too, speak of Pythagoreans,
along with followers of Anaxagoras, but in relation to the teachability
of σοφία καὶ dperd.2” Plato names Pythagoras just once, as ἡγεμὼν
παιδείας and creator of a Πυθαγόρειος τρόπος τοῦ Biov.?* It is the Society's
way of life, and not any particular knowledge, that makes his fame.
‘he doxographical sketchin the Sophistalludesto Pherecydes, Alcmaeon,
the Eleatics, Heraclitus, and Empedocles, but the ‘‘Eleatic stranger’’
obviously does not think of Pythagoras when he is listing the attempts
τὰ ὄντα διορίσασθαι πόσα TE καὶ ποῖά ἐστιν (242ς; cf. Tht. 152). Alcidamas
stated that Empedocles had studied with Pythagoras and Anaxagoras,
καὶ τοῦ μὲν THY σεμνότητα ζηλῶσαι τοῦ τε βίου καὶ τοῦ σχήματος, τοῦ δὲ τὴν
φυσιολογίαν.29 This is very doubtful as a historical statement, but it
24. Robin pronounced the final verdict arguing against Burnet, many years ago:
“que Τ᾿ ἀνάμνησις pythagoricienne peut bien étre l’origine historique de ἀνάμνησις de
Scrate et de Platon, mais que ce n’est pas un motif de transporter dans le Pythagorisme
primitif la conception philosophique qui caractérise cette derniére,” in ‘“‘Sur la doctrine
de la réminiscence,” REG 32 (1919) 451-461, at p. 455. Cf. Dodds, Irr. 209f.
2 This is attested by his contemporary Glaucus of Rhegium (D.L. 9. 38). Apollodorus
ol Cyzicus mentioned Philolaus (ibid.). Cf. also ch. II 2, n. 2.
“41K 68Boa. Against its genuineness, Zeller SBBin 1889, 996, Wellmann AbhBin
121.4, 12f, and Dicls (with hesitation); but cf. above, ch. II 3, n. 146; Frank 67; Gigon,
Ursprang 128f,—It is tempting to juxtapose Carm. aur. 12 with fr. 84 (cf. also fr. 244 and
164). Democritus praised Pythagoras (Thrasyllus ap. 1).L. 9.38).
*7.6.8 (DK TE 414.13).
ἊΝ Rep, 6ooab; above, ch. 1 4.
217.1. 8.96, from the book φυσικός; Brzoska, RET 1538. Lévy, Sources 2 n. 6, considers
the possibility of forgery; and Burnet, EGP 202, would substitute Πυθαγορείων for
ΠΠυϑαγόρου but if even Timiaeus (Cord fist Φ0 014} can anake Empedocles a direct pupil
of Pythagoras, surely a thetor like Alctdamas could have done so,
Rives an insight into the common view of Pythagoras; the “natural
science” (φυσιολογία) comes from Anaxagoras, not Pythagoras, Bug
the solemn, pompous mein which Pythagoras and Empedocles have
in common is preciscly the manner of the shaman—Empedocles the:
“deathless god” and Pythagoras the “ἢ lyperborean Apollo.”
Isocrates in the Busiris says in a general way that Pythagoras “ wag ’
the first to introduce all the other philosophy to the Greeks,” but
specifically that he “more conspicuously than others attended to
sacrifices and temple rites.”’8° It is only in the context of ὁσιότης that
Isocrates thinks of Pythagoras, not along with ἀστρολογία καὶ λογιομοὶ
καὶ γεωμετρία (23).
In his accounts of the history of philosophy, Aristotle consistently
avoids the name of Pythagoras. In the Protrepticus he describes Greek
natural philosophy as τὴν περί φύσεώς τε καὶ τῆς τοιαύτης ἀληθείαρ
φρόνησιν, οἵαν οἵ τε περὶ ᾿Αναξαγόραν καὶ Παρμενίδην etonynoavro,*
These are the living names in the field of physical philosophy ; Pythag~
oras 1s not included.*? Theophrastus names Anaximander and Xenoe
phanes as the teachers of Parmenides, and Parmenides as the teacher of
Empedocles,** where others speak of Pythagoras or Pythagoreans. If
we take all these passages together, their silence about Pythagoras the
philosopher and mathematician is an extremely remarkable coinci-
dence—if it is that.
The other tradition sets in at just about the time of Aristotle,
Aristippus dealt with Pythagoras in a book entitled Περὶ φυσιολόγων.34
Hecataeus of Abdera®® and Anticlides, who wrote about Alexander
the Great,** represent Pythagoras as introducing geometry from Egypt.
By this time a conception of Pythagoras had become dominant which
was obviously quite unknown to the pre-Platonic writers.
* Bus. 28. It is clear from Herodotus that there was some connection between Pythag-
oras and Egypt (2.81, 123), and for him too the link was in religious and cultic peculiarities,
The Busiris was probably influenced by Plato (Gnomon 33 [1961] 352).
* Arist. fr. 52 = Iam. Comm. math. sc. 79.13. The very fact that Thales and Pythagoras
are not named in this passage is an indication of the Aristotelian origin of the wording.
*? Jamblichus seems to have preserved something of this tradition in another passage
(VP τόδ, perhaps from Timaeus; cf. above, ch. II I, n. 34): Thanks to Pythagoras, he
says, the expression Magna Graecia was invented, philosophers, poets, and lawgivers
flourished there, rhetoric was born, the Gnomes of Epicharmus were much celebrated,
καὶ περὶ τῶν φυσικῶν ὅσοι τινὰ μνείαν πεποίηνται, πρῶτον ᾿Εμπεδοκλέα καὶ Παρμενίδην
τὸν ᾿Ελεάτην προφερόμενοι τυγχάνουσιν. Pythagoras himself is not named as a physical
philosopher.
*8 Theophr. Phys. op. frr. 6, 6a, 3 (on which see Dox. 477); cf. below, ch. IIE 3.
34D.L. 8.21 = A162 Giannantoni.
% Diod. 1.98.2 = FGrHist 264F25.
Ὁ D.L. 8.11 = FGrHist 140F1: cf. below, ch. VI 1.
216
Wit +e came a time
as its own Ideal of Wisdom: and there came at "
at an, who by his own innate powers has
: ‘ egtt 4
αἱ position and insight, be aa
] : scemed τὸ fulft
embodied in the persons of certain great men wi ae pied rl
the hiphest conceptions of wisdom and power |
tothe cestatic seer and priest of leaseaes) : ἊΝ ira ae
" . forerunners of Greck p
.-pot even the forerun pice!
ee w was one which the real philosophic ee
the freedom of the soul consciously
though not indeed
Pvery ape hh ,
when the ideal of the Wise M
αἰ πον a commanding spiritu
philosophe
often their point of vie , :
ὦ sclf-determination anc
toward self-determ ) oe
and decisively rejected, and ἐπα ἀρὰ ie reject,
ing.
without occasional wavering and backsliding
in reference to figures like Lameaiey and
ncient evidence
Alvis, without including Pythagoras.” a i a ΣΝ scc Pythag-
indicates that it is precisely in this perspective that heir own notion of
was, ΤῸ a later age it seemed natural to as ie ite to them
ἬΝ, “εἰ ᾿ to impt
" ‘ the past an
“wisdom” upon the great figures or ἀν eer c.” Only by such
from a modern point of view 1s “science. Thi ss of
. , This proce
reinterpretation could the ancient remain acceptable ᾿ ἐνὶ os
the renewal of ancient wisdom took place, re y> EE
Vana false start and new be
ition, with many a false a
Pythagorean tradition, ἐξ ! m the beginning,
2 ne was most characteristic of Liearperen pe Ste eRe
ic and archaic. No
' the new asthe exotic a ἄνα h
was not so much - in studying the
Appears CO testify that Pythagoras was a area ye - ane
i low he cast onlater generations, all we oe on
ae : » , ant.
πλέον τι εἰδέναι of the “shaman and hierop
Ke wrote Erwin Rohde,
hat which
M Psyche IL go = 299f Eng. ed. (tr. W. B. Hillis).
217
1. Philolaus
I. THE SPECIAL POSITION OF PHILOLAUS’ B
OOK IN
PYTHAGOREAN TRADITION ie
) g
8
To what extent the expression “
not go inte. [ἢ Any Case, we νι talki
attribute to Pythagoras and his aie
ms Stoa, ts well as popular philosophical
a - bie of material, Thesle ff, “Tents ;
2 ie Pies Thesletf, ‘Texts 1$5-U86, 243-245:
ies a Ἰ Ἰς lyponmemata, sce above, ch. 13, ἢ. }
: Me ne 4, he Lt: on the “tripartitum,' ὃ
ΚΌΠΟΥ | 468 ἢ, 2, DK 14.19.19. οἱ
ν 478 ΚΟ); also D1, ".1τ1ὸ,. Hieron εν
lato, Aristotle, and
logus 1961, and, for
van der Waerden, RE Supp. X
1. 1. 0 the ἐλ ν
, ‘ εἰπτικὰα ov
below nn, 32-31. ὑυγγράμματα,
se duane
- Posidonius (Galen Phar
‘osidon Hipp, ἢ
AR aol Clin: Mak eo
De statu an. 2.7,
=~
218
1. Philolaus’ Book in Pythagorean Tradition
disciples. There is a Platonic coloration here, and one may suspect that
this emphatic declaration served as a pretext for discarding certain
Pythagorean writings that were felt to be an embarrassment because of
their old-fashioned character.® Delatte attempted to refer the various
lines of Pythagorean verse, one of which was cited as early as Chrysip-
pus, and most of which eventually found their way into the late
compilation called Carmen aureum, to a ‘Iepés λόγος of the fifth century.®
Rostagni, still more boldly, claimed to have found the “Word of
Pythagoras” itself.” Heraclides Lembus lists a ἱερὸς λόγος in hexameters
in second position in his catalogue of Pythagoras’ writings, and, in the
manner of an ancient librarian, quotes the first line (D.L. 8.7). The
older evidence for this secret document is, however, very fragile.
Hecataeus of Abdera reports® that Pythagoras brought “the ἱερὸς
λόγος, along with mathematics and the doctrine of transmigration,
from Egypt. This does not mean, however, that he had seen a book
entitled ‘Ieps λόγος. Like the statements about mathematics and trans-
migration, that about the ἱερὸς λόγος seems to come from Herodotus,®
who does use this term in connection with Pythagoreanism. But the
context, along with the parallel passages, shows that what he is talking
about is the mythical explanation given by the Egyptians, not a “ Holy
Word” of Pythagoras° A ἱερὸς λόγος is that which is not to be
4 Plut. Numa 22, Nicom. ap. Por. VP 57: οὔτε yap αὐτοῦ Πυθαγόρου σύγγραμμα ἦν.
(This sentence is not included in the parallel report, lam. VP 252. Rohde, Q 140f, assumes
that Porphyry has inserted it. But the passage reads smoothly in Porphyry, whereas there
is unevenness in Iamblichus: the words πλὴν ὀλίγων πάνυ, p. 135.22, do not go with what
immediately precedes them, but with συνεπιλιπεῖν, line 19; and it is just here that Por-
phyry has the sentence quoted. In Iamblichus there are hints both earlier [dppyrov ... ἐν
τοῖς στήθεσιν, lines 19f | and later [p. 136.8] that there were no written works of Pythag-
oras; but, as Iamblichus repeatedly cites works of Pythagoras, he has every motive to
hush this up. Cf. above, ch. If 1, ἢ. 1§.)—David In Porph. Is., CAG XVIII 2 p. 2§.28ff,
adds Plato’s warning against written texts (Phdr. 277¢).
5 Apollonius ap. lam. VP 258, D.L. 8.7, on the ‘Jepds or Μυστικὸς λόγος. See Gigon,
Ursprung 124f.
6 Delatte, Litt. 1ff; contra, Theiler, Gnomon 2 (1926) 147ff. The material is in Thesleff,
Texts 158-163. Chrysippus, SVF I no. 1000 = Carm. aur. 54.
7 Il verbo di Pitagora. He relies especially on Ov. Met. 15; but the principal source of
that is Empedocles.
8 FGrHist 264F25 = Diod. 1.98.2 (cf. Jacoby Ila 75f). Pythagoras is named among the
sages who visited Egypt at 1.69.4 and 1.96.2. In the latter passage occurs a mention of the
ἀναγραφαὶ αἱ ἐν ταῖς ἱεραῖς βίβλοις which is regarded as a strong clue that Hecataeus is
the source. There are probably “occasional and not extensive” additions by Diodorus
(Jacoby 78.24); but the naming of Pythagoras and of the ἱερὸς λόγος is firmly rooted in
the context. Hecataeus could not avoid mentioning Pythagoras (cf. also above, ch. I
2, n. 15).
9 Hdt. 2.81; cf. above, ch. II 3—Geometry, Hdt. 2.109, metempsychosis 2.123.
10 Pace Guthrie I 160; Herodotus describes the custom, names the Greek parallels, and
remarks in closing that he is not going to reveal the explanation. Cf. ἱερὸς Adyos, 2.62;
yarallel od μοι ὅσιόν ἐστι λέγειν, 2.61; similarly 2.171, 2.51.
| } Y :
HI. PHILOLAUS
5 Ρ
ἢ 3
ἢ
in
Man 1
Εν y _ were circulated in the name of various Pythagoreans
3 Δ ] i
: ve been preserved either in fragments or entire,!* but there is
no lon 1
ie ger even any nar of their authenticity, except for some of
e fragments attributed to Phil
ilolaus and Arch
a | chytas. The Philolaus
re more important, beca
; use they treat of number th
— 3 : umber theor
an ology, while those attributed to Archytas mostly pe
na
ematics and acoustics.'4 The question of the genuineness of the
1! Above, ch. II 3. Even the scholarshi iqui
p of antiquity could not gi i
silence ani except Heraclitus fr. 129 ἐν Ion fr. pear ed
IDicacarchus (Por ὌΝ to, in this regard (above, ch. II 3, nn. 63, 52). Accordin ἣν
τς Empedaclés : δι 2 no one had exact information about the teaching of Pytha Η Ἷ
be Ne es δὰ : : cited = evidence for Pythagoreanism by Poddontis: ap ee
nymph, 8 (fr. 120) i : ἐπ}: (ἔτ. 136-137), Cic. Rep. 3.19 (fr. 135ff), Por De ant
τῶν Seca ᾿ ppol. Ref. 6.25.1 (fr. 16), Plotinus 4.8.1 (fr. 115). Porph r :
Ale boc th sina in the philosophy “of Pythagoras and Em a aes
i ae eo ihe only includes Empedocles (fr. 128, 139 = Por Pee
Empedocles by the pies 28ff) explains the difference between the two books -
Te eee one ae if that Emp. wrote the Katharmoi after his break with th
εἶ Pe eae vanes no coo felt himself bound by the vow of secrecy. In this :
ΑΘ anise actually contain, in different form, the ἱερὸς λόγος ive th ae
13 These a li ai by eae
sieished τη eg τῶν seas ᾿ 2.9217; Thesleff, Intr. 8ff; published by Thesleff, Texts;
pik Chien isa) és . Brown, Philosophorum Pythagorcorum collectionis : ecim ;
Auth. lyr. gr 133 (Lei . Carmen aureum: P. C. van der Horst (Diss. Leiden, 1932) ee ey
t. ee (Lid ἣ ΕΣ ΒΕ’ 1050) 82ff. Ocellus: Harder. Ecphantus, Diotoperies εὐ τ ι
14 This is ἃ ἐδ, ἄντ! Diels’ thod of sel ων
each eanee | : s method of selection. The amount of spuri
separated ΕἸ eee " DK, τ exceeds that of the possibly pan PUR
found in specialized aa pene ee the principle that forgeries are not likel Ἢ ἯΙ
Αἰ τας oa : ie sii and musical material. In principle this seems a :
ieee ϑ νι; ᾿ "" anes calculations like those in A16, and the ἡ ἀδιἐηξίο
scala fit aie eh a eee Eudemus (tr. 141). But there are ee
πραγματεία, the assertion that λογιστικά helps Oe nea Hired
; SCHISECUT 1 {Π0 Ὁ
1. Philolaus’ Book in Pythagorean 1 radition
Philolaus fragments has been a matter of controversy for many
decades. For A. Boeckh, who devoted a fundamental study to them,
they were “in the labyrinthine maze of the traditions about Pythag-
orean wisdom and the Pythagorean society . . .a point of light whose
radiance could, perhaps, brighten this night a little.’’25 The authority
of Zeller and Diels won acceptance for this view for a good while,
though with an important modification.1® This was true especially in
the German-speaking world; among speakers of English, Bywater 8
condemnation of the fragments, followed, with additional arguments,
by Burnet, held sway.’ Then, since Frank published his elaborate
attack against the fragments, the dominant mood has been uncertainty,
though scholarly caution has somewhat tipped the balance toward the
negative.
In spite of the mountainous bibliography, there does not yet exist
of the Philolaus fragments. But since the
ion has somewhat changed; there are new
ds, and, above all, new source material. A
with excerpts from the history of medicine
n, now makes it certain that there
a full interpretative study
days of Bywater the situat
perspectives, refined metho
papyrus published in 1893,
written by Aristotle’s pupil Meno
was a book by Philolaus extant in the middle of the fourth century
"Ὁ."
The fact that the controversy lives on, in the case of Philolaus, is in
itself significant. Writings like those of ‘‘Ocellus,” “Timaeus of
Locri,” and a great deal of the material attributed to Archytas have
matically. Archytas could determine 4/> (used in doubling the cube) geometrically,
but not arithmetically. Fr. 4 is comprehensible from the point of view of Plato’s doctrine
of ideal numbers: according to the system of derivation, number is primary as compared
to geometrical magnitudes (ἀριθμητικὴ ἀκριβεστέρα γεωμετρίας, Arist. Met. 982a26ff,
etc.). Cf. also below, ch. V I.
18 Boeckh 3.
16 The athetesis of fr. 21 (below, ch. ΠῚ 2).
17 For authenticity: Zeller I 369ff; Diels, DK 44; Reinhardt, Parm. 65 (and elsewhere;
248 n. 2 for the genuineness of fr. 20); H. Gomperz, Hermes 1932, 155 (for the genuineness
of fr. 22: cf. below, ch. ΠῚ 2); Scoon, CP 1922, GrPh 13 3ff; Rostagni, Verbo 47ff; Robin,
Pensée 6off; Carcopino, Bas. 161ff; Boyancé, Songe 99, Muses 117 n. I, 289 n. 2. Against
authenticity: V. Rose, De Aristotelis librorum ordine et auctoritate (Berlin, 1854) 2; Schaar-
schmidt (criticized by Zeller, loc. cit.); Rothenbiicher; Bywater; A. During, Gesch. d.
er. Philos. I (Leipzig, 1903) 183f; Burnet, EGP 279ff; Tannery, MSc Ill 220-243 (the
music fragments) ; Frank 139ff, 263ff; Howald, Fs. Sudhoff, Bollinger 27ff; A. Rivaud,
ed. of Pl. Tim. (Paris, 1925, Coll. Budé) 24, REG 39 (1926) 280ff, Rev. hist. philos. 3
(1929) 25; A. Lids, ed. of PI. Phib. (Paris, 1941, Coll. Budé) xxi; Festugiére, REG 1945,
1: Heidel, Maps 94; Raven, Py o3ff, in KR 308; W. Spoerri, REA 57 (1955) 278
Pres, 37 11. 140, 386E. Criticism of Frank: Mondolfo in ΖΜ 367ff = RivFil
n. $; Cherniss,
mier §72f; Moreau, Ante 14$ff; Guthrie
τις (1937) 225-245. Voubrful: Rey 183; Wuilleu
120-331: Philip 31-32.
IW A27-28 = Supp. Arist. ΠΕ 1.41, 36; εἴς below, ch, ΠΕ 2.
If, PHILOLAUS
been convicted as spurious and sct aside. The “forger” betrays himself
by letting results, or concepts, or terminology of later philosophical
thought creep in. (An important criterion here is the a “of
Aristotle that among the Pythagoreans the distinctions had n ee
been fully developed between form and matter, sensible and st
insensible, corporeal and incorporeal.)!® ἀπο δι sign of for ne :
close an adherence to the wording of a passage of Plato ge : ᾿ :
even when Plato is dependent on Pythagoreans, he does not co we ᾿
for word. According to these criteria the metaphysical nee
Archytas,” for example, are unquestionably apocryphal,?° but th
case of Philolaus cannot be settled in this way. — μω
It would be convenient if external, linguistic factors could b
decisive. Burnet tried to show that a Pythagorean of the fifth cent
even in south Italy, would have had to write not Doric but tonic
Greek, the dialect that had become standard for philosophic i
scientific writing. The physicians of Cos and Cnidus 3 toni ᾿
did Antiochus of Syracuse! and Herodotus, deserting in each ᾿ he
dialects of their home cities.2? Burnet will admit that Archyt een
Doric,” as Thucydides chose Attic. But the author of i ἅδον, ᾿
Xenophontic Constitution of the Athenians was already sere x 2
why would Philolaus have had to restrict himself to Ionic? The ond
τετρακτύς perhaps points in the direction of Doric. But, aleve all,
19 Above, ch. I 2.
20 E.g., there is Aristoteli
8.» totelian terminology i ee,
Fs ele SPS y in Περὶ ἀρχῶν, p. τ jn He
Oo ns τῶν τοῦ » P. 19.19 Thesleff:
fe aaaeds a eee τόδε τι εἶμεν, ἃ δὲ Wola τὸ ὑποκείμενον, παραδεχόμενον τὰν wk ad
Rep. so9d Ὁ na The | ea pues) is mostly copied word for word ἍΜΑ ἢ Η
setae ye : ogical writings of ps.-Archytas (pp. 15-19, 21-32 Thesleff ) we
sane ᾿ ristotelian that even Themistius supposed they were written not b τὴ
bi i6a cia a by a later Peripatetic named Archytas (Boeth. In Cat. I ee on
. > . . t , 1 , Ἑ οὐδέν.
like option of Ps gs zs Aaa, ov μάλιστα καὶ γνήσια λέγεται εἶναι τὰ συγγρά θυ
a νου nie 4s a - P. $6) and since he is somewhat later than Philolaus, is woe d
oe εἶμι ἀν, 7 by ἐν τας It is easy to overlook the fact that more as ἐπ παι
in that of any other Pythago i .
heseworkvare ot printed DEK ythagorean, since the rather extensive fragments of
21 FGrHist §55F2. In Al 5
. cmaeon’s fr. 1 are to be found :
lonian Κροτωνιή ound not only a Doric ἔχοντι b
Ans. Burnet (EGP 28 xov7e Dut an
Cie: ( 2n. 45) and Thesleff (Intr. 80) believe that he wrote in
22 The “ oo 7
(ch. ἢ x are attr ibuted to Pythagoras are Ionic (D.L. 8.6ff), as are “Androcydes”
cai ee ᾿ as ον 4.25.50, 4.28.19; pp. 142-145 Thesleff). It is eee
; at, as many assume, the so-calle Saas : oo ere
Isis ἀπ ΠΥ ραν Uled Hypomnemata were Tonic; see Burkert, Philo~
"8 EGP 283,
44°The kappa i ae a:
Peers aN ee Is surprising, compared with Ionic χιλιαστύς and Lesbia
% not simply “Dorie.” Schwyzer (I ἜΘΕΙ
cf. ' Pees eerie” yzer ([ 407) sugpests analogy wi Se
in Poni, Delian Inscriptions, τρίκτοια in Sophron fr. 7 Kaibel es I 7. oaks
Ἵ Ε ‘ ' 1 Ε ᾿ « . ἔν ΐ
Cael. «1.1 ον Delphi. Ια ἰν doubtful whether the form: Ζανὸς πύργον (Simpl,
“οἰ, §t2.t2- Arist. fr. aog, γος, da uel. go.7: y _ ces ae
sea Qe ΤΩ " ' ᾿ ᾿ ᾿ ᾿ ‘ 74 ν.1. 7: μός Cath be “a: aes
see Schwyvzer 1677 n, 4. Schwyzer, Dial. 696 (Chios), pds) ¢ counted as Doric;
1. Philolaus’ Book in Pythagorean Tradition
are told that the physician Acron of Acragas, whose floruit was about
430 B.C., wrote in Doric;** and it is beyond question that Corax and
Tisias spoke and wrote in Doric. All this shows that prose literature in
Doric began to appear at Icast a generation before Archytas. The indi-
vidual dialect forms, however, are so often inconsistently reproduced
in the manuscript tradition that even obviously false forms do not
provide a criterion of spuriousness. Hyperdorisms are to be found in
Sophron, Theocritus, and Archimedes, and may even have penetrated
into popular speech.”°
Gencrally speaking, definite proof cannot be offered for the authen-
ticity, but only for the spuriousness, of a book. But the defender of a
book’s genuineness 1s not confined to destroying the opponent's
arguments, knowing the while that new ones may always appear.
What one can and must do is to ask where, in the transmission, the
sources of error lie. ‘‘Forgeries” are usually pieces of quasi-historical
reconstruction; the Pythagorean pseudepigrapha, for example, show
what people wanted to be regarded as Pythagorean. We can detect in
them a certain tendency of interpretation, a general purpose which is
also discernible in the distortions of the doxographical tradition. A
genuine fragment must show itself so by standing aloof from this
tendency and not being deducible from it. In this way one can establish
the presumption that something is probably genuine.
First let us look into the external evidence for Philolaus’ book. In
various ways, Philolaus’ name is connected with the beginnings of
written Pythagorean literature.2” According to a tradition that gocs
back at least as far as Satyrus, Plato in a letter to Dion commissioned
him to buy “from Philolaus”’ three “Pythagorean books” for the
τῳ. There is no reason to think of a pseudepigraphon
(pace Thesleff, Texts 1). The doctors in comedy speak Doric, from Crates (fr. 41 Kock)
onward, see Euphron fr. 3, Alexis fr. 142, Menander Aspis 439-464: one of them is said to
come Σικελᾶς ἀπὸ yas, Epicrates fr. 11.27. The cookbook of Mithaecus (Pl. Gorg. 518b)
was also Doric (Ath. 7.325f).
26 Cf. Schwyzer I 185, 719 n. 6. On the dialect of the Pythagorean pseudepigrapha,
sce A. Matthaci, De dialecto Pythagoreorum (Géttingen, 1878); R. Fohalle, “La langue
d’un texte ‘doricn’” (Aisara) in Ltrennes de linguistique off. ἃ E. Benveniste (Paris, 1928)
27-49; Delatte, Pol. 75ff (“Archytas”’ Περὶ νόμου καὶ δικαιοσύνης), 127“ Hippodamus’’);
L. Delatte, Les traités de la royauté d’icphante, Diotogene et Sthénidas (Lige, 1942); Fi.
Wilhelm, RhM 70 (1915) 161-223 (Bryson, * Callicratidas,” ‘ Perictione,” δ Phintys’’);
Thesleff, Initr. 85-92.
27 Wiersma correctly analyzed the tradition, Muaemosyue 1942, 23f. ‘The mistunder-
standing about the “three books of Philolaus” hung on, from the tine of Boeckh, until
it. made its way into DK, in spite of the express testimony γέγραφε δὲ βιβλίων ἕν (1}.1..
.8.4). Bywater (26fF) and Schaarachinidt (7411) capitalized on the contradiction,
25 Suda s.v. "Ακρων: Δωρίδι διαλέκ
Hl, PHILOLAUS
stupendous sum of 100 minas; Philolaus had “made these book
known” (ἐξήνεγκε).28. This alleged letter of Plato must actuall
existed, to judge by the form of citation;2* and its purpose is ne hard
to gucss. Ancient publishers frequently preface apocryphal publicati
with a letter that gives an account of the origin and importance of Ἧ
writing.*° Thus there was in circulation, at the time of Satyrus abo ;
200 B.C., ἃ set of three pseudonymous Pythagorean books erode rd
by this letter from “Plato,” which named Philolaus hou h not
author. The word ἐξήνεγκε is used of someone lic “bei, ς out” ᾿
publishes a writing previously unknown, perhaps secret - in :
case already in existence for some time; but it is not used of the a i.
himself.#! We must get rid of the misconception, which goes Ὁ ᾿ 2
Bocckh, that Philolaus had written the “three ἜΒΗ ἐν 7 iy τὰ
will we be able to identify them. ἀμ
Σ θ β β θ ᾽ ( )
D.L. 8.15:
ee Iam. VP 199:
é
éxypt τε Φ Ἵ ) D ei
Hs ia ies ον οὐκ: ἦν “a γνῶναι οὐθεὶς οὐθενὶ φαίνεται τῶν Πυθαγορείων
μα: οὗτος δὲ μόνος ὑπο Ἵ
ΕΠ , : μνημάτων περιτετευχὼ ) τῇ
sk XYos πρὸ τ
ie 2 oe τρία βιβλία, ἃ Φιλολάου ἡλικίας" ἀλλ᾽ οὗτος Ὁ 18
peat πέστειλεν ἑκατὸν μνῶν ἐξήνεγκε τὰ θρυλούμενα ταῦτα τρία
: βιβλία ἃ λέ f }
β ς έγεται. Δίων ὁ Συρακούσιος
exatov μνῶν πρίασθαι Πλάτωνος
Bee κελεύσαντος.
Fe incre eee (Φιλολάου) Πλάτων ὠνήσασθαι τὰ βιβλία τὰ Πυθαγορικὰ
ee . « ᾿ t
Fie oe a ᾧ : ae (Platonem) tris Philolai Pythagorici libros decem milibus
i um. enarius = drachma, so th il
Αδαδ ae τ us = » So that 10,000 denarii = 100 minas.
perce ie (Platonem) Philolai commentarios esse nanctum.” Tzetzes Saat
ous Soran! of Satyrus, Hermippus, and the letter of Lysis to produce :
. i ᾿
ἐμ eu τ fhe Dion’s help, Plato bought a book of Philolaus, for 100 ae
poverished Pythagorean widows (Chil. 10.790ff; 11 208). Cf. also th
Prolegomena to Plato 27, p. 201 Hermann _ ena
29 4 1
αφει 1D.L. 8. ler i i
ἜΝ fine ae saa ea is ee about this (Gnomon 2 [1926] 587), as is Beutler
E : ers argument, that the letter would ἢ if i
εὐ ΕΝ are uld have been famous if it h
i ae is not persuasive; even the OpvAovpeva τρία βιβλία perished almost Na
shi = = Se sources give various, often contradictory, guesses as to where
a ae : ae ina Sri ap. D.L. 3.9 says Dionysius gave it to him; Gellius
3.17.1 5. according to “certain persons.’’) Al
ἀρ κεν ion, Ρ 5... Also, how could ἃ Pythagore
is ᾿ a sri man? (lam. VP 199 says Philolaus was in need: erie ἔπε οι a
. . ‘ 4 i iy i
nae sda eet oe ee by certain Pythagoreans in gratitude for his
havin ‘lease from jail of one of their numbe
εν circumstances were devised to fit the “fact.” Pore ear
C : ; ‘ “,
ie fran tat 1961. Examples are the letter of Lysis and the Hypomnemata
atte, Litt, 1036), the correspondence between PI
at ! ato and Archytas and Oce
Oe Macc., Diogenes Antonius, and Dictys Cretensis : a
ἐξενεγκεῖν is used of Hippasus in the versi a
veyKe asus ersion of the mathematic, and in the same
ἐξενεχθῆναι is used of geometry (lam. VP 88 Comm. math, se | " ieee
, Js. p. 77.19). The abbreviated
expression “Philolat Jibr
pression Philolai libros, comentarios” in Cicero and Gellius is irrele
question. reiius is irrelevant to. this
1. Philolaus’ Book in Pythagorean Tradition
“Pythagoras wrote three books: On Education, On Statesmanship,
and On Nature.”3? This so-called tripartitum was available to Heraclides
Lembus, who also used Satyrus,?° and also to the author who put
_ together the account which is at the base of the lives of Pythagoras in
Diogenes Laertius and Hesychius.*" For a time it was regarded as just
what it claimed to be: the authoritative and genuine work of Pythag-
oras.
It was natural enough to invent a correspondence between Plato and
Dion about Pythagorean writings, just as the author of the “ Ocellus”
book represented Archytas as corresponding with Plato. The remarkable
thing is that Philolaus was included in the plot, in spite of the chrono-
logical difficulties involved. Apparently this was the result of the
belief, attested at least since Neanthes, that before Philolaus there were
no Pythagorean writings known. No anecdotal or legendary features
characterized the publication of Philolaus’ book.** We have only the
simple datum: Philolaus and the first appearance of Pythagorean
writings belong together. |
We have from Hermippus another story, older than that of the
“tripartitum”: “Ης (Philolaus) wrote one book, which according to
Hermippus some writer said Plato the philosopher ... bought from
the relatives of Philolaus for forty Alexandrine minas of silver, and
from which he copied his Timaeus” (D.L. 8.85). In spite of similarities,
32 D.L. 8.6; cf. Schol. Pl. Rep. δοοῦ, and the Suda s.v. Pythagoras (from Hesychius).
33 On the “tripartitum,” Diels, AGP 1890; Lévy, Sources πο, Thesleff, Texts 170-172.
Heraclides Lembus ap. D.L. 8.44 is dependent on the ‘“‘tripartitum” citation at 8.10.
34 Cf. nn. 32, 35. (Was it Neanthes?) Further, above, ch. I 1 ἢ. 22. The obvious identifi-
cation of the “three books” mentioned in the supposed Platonic letter with the “ triparti-
tum’ (made by Wilamowitz, Platon II 87; Wiersma, Mnemosyne 1942, 23) has often been
prevented by the prestige of Boeckh. In his desire to fit all the data on Philolaus into a
single harmonious picture, he decided that Philolaus had written a work in three books
(18ff). This led Diels to the opinion that the two three-book works were in “competition”
(AGP 1890, 461f). Harder (150) connected the “Plato” letter with Philolaus Περὶ ψυχᾶς
(ἔτ. 21).
36 Neanthes FGrHist 84F26 = D.L. 8.55: μέχρι Φιλολάου καὶ Ἐμπεδοκλέους ἐκοινώνουν
οἱ Πυθαγορικοὶ τῶν λόγων; the continuation, with its account of the break between Emped-
ocles and the Pythagoreans, is connected with Timaeus FGrHist 566F14 = D.L. 8.54, on
the λογοκλοπία of Empedocles (cf. κωλυθῆναι in Neanthes with τῶν λόγων ἐκωλύθη
μετέχειν in Timaeus). There is nothing further in this passage about Philolaus, but D.L.
8.15 would be a natural continuation: οὗτος δὲ μόνος ἐξήνεγκε τὰ διαβόητα τρία βιβλία.
1 ροποϑ (8.15) and Tamblichus (VP 199) emphasize that no Pythagorean writings were
known before the time of Philolaus. Demetrius of Magnesia attests that he was the first
Pythagorean to write a book (1).L. 8.85); see also Euseb. Adv. Hierocl. 11, p. 380 Kayser
(IDK 44A8).
M6 Tamblichus’ statement that Philolaus was impoverished (above, n. 2y) is related to the
Lysis letter (Hercher, Lpistologr. gr. p. 603) and to the version of the ἐξενεγκεῖν of geometry
found at lam. VP 8g Conan. maflt. se. p. Lene
Il. PHILOLAUS
this report is plainly different from those previously mentioned. Onl
= book is pacman it was written by Philolaus himself, onl εἰς
Pr Νὰ — Raaes — mention of Dion or a letter from Plato,
7 | introduction of a forgery, but a charge
against Plato, that the Timaeus was a plagiarism from Philolau ᾿
"εν a ae of are is to be dated oe eG
atyrus, and himself cites a still earlier
᾿ the ae also shows that this story is earlier a. iecenae
egend.®° The latter presupposes, without explanation offered, that
could get Pythagorean books from Philolaus. In this stor | ae
connection is alleged between Plato and Philolaus’ book, λει ἘΝ aa
theme is present in a still older pi
er piece of evidence. Ti
erapher, addressed Plato with the reproach a
ee silver hast thou spent for one small book
rom which thou then Timaeus learnd’st to write.?®
This i
ς ee : be taken as meaning that Timon was the inventor of the
: ’ a
a Ν : ae of we plagiarism are plentiful and even earlier
is. Alcimus found Plato’s phil isting J
osophy preexisting in Epich
—= hilc g picharmus;
ding to Theopompus, Plato imitated Aristippus and Bryson in his
8? The expression ‘‘Alex i
andrian minas”’ is an ee
21, S : ΡΝ ἢ ᾿ appalling historical bl
city ἫΝ aa a Ν rain that an Alexandrian writer did a ee ies
y Alexander the Great. Perha . at his
: ᾿ Ῥ5 ἃ conve i :
and has been lost. (Alexandrian money was famous: ‘DL ie μον cea
Inscriptions of Cos 34.) : D.L. 7.18; Paton and Hicks,
38 The “‘tripartitum”’ le
gend drove the other one :
100 mi - out: three books 1
eae na of μι ΕΣ ΠΕ τὸ himself the author. Though ἼΣ λυ τὰ ΜΝ ΗΝ
᾿ a work of physical philosophy, 1 6 pes . ” ποὺς -
foregound, in true Hellenistic fashion (Diels, ACP ae a anette
39 a + > ; oy :
aes 5 ἀργυρίων ὀλίγην ἠλλάξαο βίβλον,
ἔνθεν ἀπαρχόμενος τιμαιογραφεῖν ἐδιδάχθης
Timon fr. PPF) = :
aa Aaa A a re νὴ == Gell. 3.17.6; also Procl. In Tim. 1.1.11 et sae
Scien koe this snes — ee be the συγγραφεύς mentioned by ee
Seite akan ay Aa designates a prose writer (Pl. Phdr. 278e, cf
7. Aristoxenus is regarded h per 786, cf. 2350), pace
EGP 270f: Wi : δ as the originator of the allegati
sen he Update, vee ᾿ 87; Geffcken [below, n. 41] 94; Wier disenat a
’ ’ 306; arder 41) Wehrli A ἢ ᾽ ᾿ ; Υγ56Ὲ
the Hermi ΤΕ ᾿ ‘li (Aristox. 67) does not distingui
εὐόνει mpees Heme se that from Satyrus, and arbitrarily πο: pret
enus fr. 43, as though the latter Id 1 | pie ta
The fact that a citati fans could have known the διαβό ᾿ ᾿
ee : ὩΣ ἃ citation from Aristoxenus follows proves nothing ees ee aden
ἘΝῚ ὍΣ ς anethods of composition. lam. VP τοῦ (DK 14.07) wands he Reh melopenes
per aoa (lam. F Ρ 1y7f Αὐἰδίοκχ. fr. 10; lam, VP 205 ff pre ter cies
ie ᾿ a ee bepins at lam. ΚΡ 200; cf DK $818) ee ᾿ τὸ ΓΕ
᾿ hi ul Peed Ἷ a : ω ‘ ἠ ynte ΜῈ
ΤᾺ ἀν τὰ ἡ ἢ 40 ee by 31) belongs to the book Περὶ ΑΘ ἀμ Bl ᾿ ἬΝ
ἀπ κω tree, ena bead, lamblichus ases a mosaic technique, too The sae
' iheod that the author may be Aratosca ἫΝ . ees
ἢ ᾿ IN, lout sere hainagy approachin ‘
: g certainty.
1. Philolaus’ Book in Pythagorean 1 radition
dialogues; Aristoxenus stated that the Republic was contained σχεδὸν
ὅλη in the ᾿Αντιλογικά of Protagoras;* and anyone who wanted to go
back still further could maintain that Plato “stole” the doctrine of
immortality from Homer.
It is a mistake to suppose that the originals of Plato's alleged pla-
giarisms were ad hoc forgeries. Not only the Homeric poems but the
books of Protagoras, Aristippus, and Bryson were readily available;
and even the Epicharmus fragments cited by Alcimus may be genuine.*
The charge of plagiarism 1s like a philological discovery, ἃ clever
inspiration that is all the more effective if the similarity is not apparent
on the surface. The keen-witted Kritiker detects the “theft”? which
completely deceives the man in the street. Therefore we must suppose
that the book of Philolaus, too, upon which the Timaeus was said to
be based, really did exist, but not that it showed so thorough an agree-
ment as does the “ Timaeus Locrus” book. It is not necessary for the
similarities to be much closer than those between Homer and Plato,
for the malicious assertion to be made.
The conclusion will be, then, that before Timon, in the fourth
century, there was in existence a book of Philolaus which could
plausibly be brought into connection with the Timaeus. The later
tradition, and especially its use in the story of the “Tripartitum,
shows that this book was regarded as the oldest book by a Pythagorean.
What is more, the Menon papyrus proves the availability of a book
of Philolaus in the fourth century.” According to this witness Philolaus’
book contained discussion of medical matters; and such topics take
up considerable space in the Timaeus.
Scholars have assumed that this book, though relatively carly,
cannot have been written by Philolaus, but must have been com posed
by someone else and only later given the prestige of his famous name.
But this leads immediately to the guestion, how famous he was and
41 On these accusations of plagiarism, sce Zeller I 1, 429 0. 73 E. Stemplinger, Das
Plagiat in der griechischen Literatur (Leipzig, 1912) 25ff; J. Geffcken, “ Antiplatonica,”
Hermes 64 (1929) 87-109. Alcimus: D.L. 3-9-17 - EGrHist §60F6 DK 2301-6;
Theopompus: FGrHist 115F259 = Ath. 11,508c-d; Aristox. fr, 67 == D.L. 3.37) 57’
42 Ath. 11.597¢ (from Herodicus).
48 Alcimus forged the Epicharmus quotations he used, accordin
Covotti, 1 Presocratict (Naples, 1934) 144. Bywater (26.1) cast d
book, though it is in Diogenes’ catalogue of his works (0.55).
44 Therefore it is not necessary to assume that Timon had the ‘Timacus Locrus”
book or the like, as Proclus thought (above, n. 39):
45. Α27-.28. “Tt must be based on something in writing, | Says Wilamowitz (Platon Π 88).
40 Below, n. 86.
g to the view of A.
oubt on Protagoras’
Hl, PHILOLAUS
what he was famous for. Practically nothing is known of his life.4?
His home was Croton, or maybe Tarentum, and he spent some an
in Thebes—all the rest, what little there is of it, is demonstrabl
embellishment or simple misunderstanding.** The ἜΘΗ of Philola
in the Phaedo provides one fixed point chronologically: he must ἘΝ
been in Thebes before 399, and was therefore a contemporary of
Socrates, though perhaps somewhat younger. Another fixed oS
can be found in the statement of Aristoxenus?® that the “‘last P tha -
orcans, with whom he was acquainted, had been pupils of phic
and Eurytus. Archytas spoke of Eurytus; and Philolaus, Eurytus, and
Archytas are frequently associated in the tradition.5 Roald of
“3 : ἱ
Νι Wai Platon 1 86f; Frank 294 n. 1 (hypercritical: “timeless, mythical
δ 5 Wuilleumier 5668. The statement of Raven (KR 312), that ‘‘there 4 abund
information concerning Philolaus in the works of several later writers,” i cae
pares iters,’’ is at best mis-
48
as Menai is ἰὼν Oe πὰ ἢ ἘΠῚ Tarentum: Iam. VP p. 144.11, In Nic
I, I. 44A6), Claud. Mam. De statu an. 2 |
. .2.3 (DK 44B22
δ χαμὸ τὰ scans aie aes ae Wuilleumier (567) tries for pesca i i
n, and lived after the catastrophe in Tarentum.—Pl
.—Plut. De gen. 13.
" ἐν pasar eae ᾿ the dialogue, represents a Pythagorean cael Eos
ome to Thebes for the purpose of making offerings inh
i ee δ τα g offerings in honor of Lysis. He relates
Ξ : ped the great catastrophe of the Pyth
with Philolaus. As against the re i Lac aS νρς
port of Aristoxenus, who calls Lysis’
(fr. 18, followed by Neanthes FGrHi see ae aan
: rHist 84F30), Plutarch’s novelistic trea
, tment h
pent ae aaa in place of the completely unknown Avaepus
rus (In 13, p. 9 Norvin, followed b :
: Pe ; y Schol. Pl. Phd. 61d =
“51 ὧν Philolaus himself traveling to Thebes and says that he escaped es ae
eae us (= i He has apparently combined elements from Plato Plutarch
et nus, ἢ the Lysis letter.—The statements about the poverty of Philolaus ate
1 ions ea out the story (Iam. VP 199; above, nn. 29, 36).—According to Synesius
repre te > 134 Terzaghi; not in DK; followed by Theophylactus, DK I ἐν 19)
nd Archytas were generals. This seems to go back -
(De or. 3.139) which asserted that Phi ioe eae pee yea
ilolaus the philosopher h
the general.—The report that Philol ee ee
aus was killed because he was believed imi
at tyranny is based on a misunderstanding of hi ooaees τὰ ae
Aine armeae 3 g is source by Diogenes Laertius (8.84;
: 567; Maddalena 335); the source referred i
mentioned just before.—The account of a ti cree es ties
, ‘fore.— meeting between Plato and Philol
eas be an invention based on the story of Plato’s ici ace eens
2 ei of fame (VH 1.23), ἐν τοῖς Ἕλλησι τοῖς πάλαι μακρᾷ τῇ δόξῃ διέπ one
: ee saa sia ag Bie Πρωταγόρας Δημοκρίτου, τῇ δὲ oodia ἀράν
, ὅσον ἀνδρῶν παῖδες. This si ἱ ς᾽
οὐ ὼς τὰν are ς is simply means that Philolaus’ fame came late—
Fr. 19 -- ID.L. 8.46; above, ch. II ς.
πεν ΠΝ oe in the catalogue Iam. VP 267, p. 144.11; as immediate pupils of Pythag
as, ibid. 104 (see ἢ. 41); Philolaus and Euryeus i ‘ctl ΨΥ
Eurytus as ἃ pupil of Philolaus in th ee
aus ¢ legend, Iam. VP 139, 148 (related ἢ
and τὸ what Hermippus tells of Pyth: Δ ΠΕ ae ἐν λλθδ
s tells ythagoras and Calliphon, Joseph. A
! ΜΕ ΕΣΗΝ " i » Joseph, Ap. 1.164 --- DK 19.2);
eae as teac a of Archytas, Cic. De or. 3.139 (the report of seer oe Fat 7
y mentions Archytas, cf Harder 43 0.1); Vitr. 1.1.16, Synesius (above, ἢ, 48) ie
. ᾿ a ' , “
1. Philolaus’ Book in Pythagorean ‘Tradition
Cyzicus, who must be dated earlier than Epicurus, said that Philolaus
was teacher of Democritus.** Philolaus is, then, a name that appears
Ὦ accounts of teacher-student relationships, which is connected with
the Pythagoreans of Archytas’ circle, of Phlius, and of Thebes, but
which is not associated with a body of legend or anecdote. In the
Phaedo, Plato represents him as having spoken about the prohibition
of suicide, but as not having said “anything precise” about the reason
for it.22 Aristotle cites a saying of his,” but it has rightly been emphasized
that nothing in Plato or Aristotle would give reason to think that he
had written a philosophical book;” and apparently the ancient com-
mentators on the Phaedo could find nothing relevant to their purpose
in. a book of Philolaus.®* The meaning of all this is that the tradition
of the special role of Philolaus as the first Pythagorean who published
writings, and in particular of his special relation to the Timaeus,
cannot have been spun out of Plato and Aristotle,5* but must have been
based on a fact independent of them—on the existence of a book by
Philolaus in the fourth century B.c.; and we know from Menon that
there was one. Philolaus’ special position resulted from the fact that
this book was the first—and for a certain period, perhaps, the only—
written exposition of Pythagorean speculation on nature and on num-
ber. Naturally, we shall attribute to this book, and to Theophrastus’
use of it, the doxographical reports pertaining to Philolaus.
51 D.L. 9.38 = DK 74.2 = 44A2. Apollodorus (v.l. Apollodotus) 15 dated from his
connection with Hecataeus of Abdera and Nausiphanes, who is reported to have been
teacher of Epicurus (Clem. Al. Strom. 2.130; DK 73, 74, 75).—A letter purporting to be
addressed by Pythagoras’ son Telauges to Philolaus, and mentioning Empedocles, was
already pronounced spurious by Neanthes (FGrHist 84F26 = D.L. 8.55, cf. 8.53, 8.74).
Cf. Nicom. Ench. 9, p. 252-13: Φιλόλαος ὁ Πυθαγόρου διάδοχος.
55 P]. Phd. 61de. Suicide was regarded as dishonorable in Thebes (Arist. fr. 502).
ὅ8 FE. 1225230 = DK 44B16; cf. above, ch. Il 4, n. 139.
54 Frank 294, Raven in KR 310, 312.
55 To what extent that which follows in Plato—the φρουρά motif and the thought
that a human being is ἕν τῶν κτημάτων τοῖς θεοῖς is to be attributed to Philolaus is
uncertain. In any case Xenocrates, in his statement that the φρουρά is Τιτανική καὶ εἰς
Διόνυσον ἀποκορυφοῦται (fr. 20; see Heinze 150ff), is dependent on Orphism (Pl. Craft.
4ooc) and not on Philolaus. Nor are the other commentators, led by Olympiodorus
(In Phd. p. 84f Norvin); therefore there was nothing on this subject in Philolaus’ book.—
Olympiodorus’ manner of citing acusmata to explain the phrase οὐδὲν σαφές (In Phd.
A 113, pp. 8f Norvin) shows that he is thinking of them as allegorical, and therefore he
is drawing on a late tradition, not Philolaus. Cf. above, ch. "ἢ 4.
δ Το be sure, a book was attributed to Cebes, too—Cebetis Tabula (see T. Sinko,
“De lincamentis platonicis in Cebetis q.v. tabula,” Eos 45 [1951] 3-31; he dates it in the
2nd century A.p.). Anda whole series of writings, of the most dubious authenticity, are
attributed to Simmias (1).L. 2.124f; ¢f. Hobein, RE MTA 144-155).
Π. PHILOLAUS
From the time of Speusippus, Xcnocrates, and Heraclides, a Pla-
tonizing interpretation took the place of what Aristotle gives us as
the Pythagorean tradition; and in the succeeding era it held the
field almost exclusively. Pythagorcanism no longer seemed meaningful
except in this form, so that it was propagated by the Pythagorean
pseudepigrapha and taken over by the neo-Pythagoreans. Only on
certain special questions did one consult Aristotle; in general, the
Platonic “system of derivation” was supreme, and there was aS
polemic, direct and indirect, against Aristotle. One exception stood
contrary to this powerful and, for the most part, unified tradition—
Philolaus.
According to the explicit testimony of Aristotle, Plato replaced the
opposition of πέρας and ἄπειρον which had figured in Pythagorean
speculation, with the pair ἕν -ἀόριστος Suds, so that the place of older
cosmological ideas is taken by a system of ontological derivation, whose
leading ideas are in the realm of the categories form-matter ies
passive.®” This system Speusippus regarded as Pythagorean, and in
Aétius too it is presented under the lemma “Pythagoras.” Nothing is
said of πέρας and ἄπειρον. In the enumeration of Opinions about the
ἀρχαί, however, we read: Φιλόλαος ὁ Πυθαγόρειος τὸ πέρας καὶ τὸ
ἄπειρον. 58 When, later on, the opposites πέρας and ἄπειρον enter the
scene once more, the earliest reference, Nicomachus, adds a quotation
from Philolaus.®° In this fundamentally important line of thought
in which, from the time of Theophrastus, Platonic terminology ie
content had replaced the pre-Platonic Pythagoreanism that Plato and
Aristotle themselves discussed, the name of Philolaus appears as the
sole representative of the original Pythagoreanism.®°
Nor does this observation stand alone. The most fundamental
difference between Pythagoreanism and Platonism, according to
Aristotle, is χωρισμός, the discovery and definition of a supersensible
incorporeal realm of being, of which the Pythagoreans knew dee
Theano,” Syrianus, and Proclus contradict Aristotle on this point.®
7 Cf. ch. I 2-3.
δ A&t. 1.3.10 = IDK 44Ag.
: Dea Ar. 2.18.4, citing Philolaus fr. 2; cf. above, ch. I 3, n. 43.
The difference struck so late an author as Damascius. On the one hand, he k
the doctrine of Pythagoras,” about the Monad and the Indefinite D tran ἘΠῚ
by a supreme Onc (Princ, [yi.20 Ruclle; cf. Eudorus, above, ch. 1 3, ἐὰν ἡ) Bite ee =
ἡ τ ee een al ited doctrine of the Platonic Philebus, which is connected with
ilolaus frei (1 rine. Ptor3; 1 αα τὰ Ruelle). Thus the doctrine of “Pythagoras” and
that of the Platonic Philebus are seen as contrasted (1 86.20; 1 ΟΝ, 1 ΗΕ Ruclle) ""
" Above, ch. 12, nn. 22-23, 13, m2. ᾿ |
1. Philolaus’ Book in Pythagorean Tradition
Along with the Platonic development of a hierarchy of being went a
terminological differentiation between ἕν and μονάς, though in two
different ways. While Speusippus always called the highest principle,
beyond being, ἕν and distinguished it from the μονάς in the realm of
numbers,® the “Pythagorean” tradition set up the povds as the basic
principle and relegated the ἕν to the realm of number and of sense
perception.® All shared the objective of establishing clear terminologi-
cal distinction between the realms of existence. But Theo adds, in the
passage in which he deals with the distinction of ἕν and μονάς, ᾿Αρχύτας
δὲ καὶ Φιλόλαος ἀδιαφόρως τὸ ἕν καὶ μονάδα καλοῦσι καὶ τὴν μονάδα ev.54
In Philolaus and Archytas he could find no warrant for the separation
of levels of being as the Platonists thought of them. Actually, according
to Aristotle the Pythagorcans uttered the words μονὰς καὶ ἕν in a
single breath.** Once more we see Philolaus, with Archytas, standing
apart from the Platonizing line of interpretation.
Even more striking is a similar situation in the tradition about
the movement of the earth. According to Aristotle, the Pythagoreans
believed there was a “central fire,” about which revolved, first, an
invisible “counter-earth,” and then our earth, “as one of the stars,”
and then in successive paths the moon, the sun, the five planets, and the
fixed stars. This astronomical system, which has been celebrated in
modern times as an anticipation of Copernicus, disappeared, except for
a few traces, in the later Pythagorean tradition. (The ancient debate
over the word ἐλλομένην, which Plato uses of the earth in the Timaeus,®
kept alive the idea of the earth’s moving.) According to the
Hypomnemata (25) the cosmos has the earth ‘‘in the middle,” and the
same thing is presupposed by the biography in Photius®’ and by
6 Cf above, ch. I 3, n. 62, lam. Comm. math. sc. p. 17.15 (for the attribution to Speu-
sippus, see Merlan, PiNeop! 96-128). Xenocrates, according to Aétius 1.7.30 (fr. 15 H.),
spoke of the Movds as the highest god, and according to Favonius 5.7, of the ἕν (unum)
as the highest principle (fr. 16 H). Perhaps it was not until after Speusippus and Xeno-
crates that terminological exactitude came to be highly valued.
68 Above, ch. I 3, n. 30. Also ps.-Archytas p. 47.29 Thesleff.
64 Theo Sm. 20.19f = DK 44A10 = 47A20.
86 Arist. fr. 203 (p. 139 Ross) = Alex. Met. 39.15: τὸν νοῦν μονάδα τε καὶ ἕν ἔλεγον.
66 Below, ch. IV 2, n. τό (Tim. 40b).—The counter-earth is used in the interpretation
of the Timaeus by Chalcid. In Tim. 122; further, Plut. Numa 11, De an. procr. 1028b.
Por. VP 31 is unclear (dvrixOwv . . . ὑπὲρ ἡμᾶς). Nicomachus Th. ar. 59.5ff, 82.3, 8 (10
σφαῖραι, the moon the third from the bottom), Lydus Mens. 4.51 p. 108.5.
67 Anon. Phot. 4290 178: under the 8 spheres of the heavenly bodies, the spheres of the
4 elements fire, air, water, and earth (cf. the Stoic teaching, D.L. 7.137, 155, ps.-Arist.
De mundo 392a32ff). Asclepius (Met. 35.19ff) counts the elements below the moon as one
sphere and adds the counter-carth as a tenth, in an attempt to reconcile the “10 spheres”
with the geocentric system. (Perhaps the same motivation is present in Por. VP 31 and
Th. ar. $9.5f, 82.3, 8; cf. also Origen, comm, on John, 13.40.266).
Ill, PHILOLAUS
Aétius.°* Theo quotes as Pythagorean doctrine the verses of Alexander
of Ephesus on the harmony of the spheres, in which the earth is ex
pressly said to be at rest in the center of things,®® and Timaeus Locr :
simply replaces Plato’s worrisome ἐλλομένην with the word ἱδρυμένα 7
There is even outright polemic against Aristotle’s report. In the
ancients’ way of thinking, diametrically opposed to the modern in this
point, the idea of the earth’s moving was a false notion which had be
scientifically refuted ;?! and this forced upon them the task of ioe
that the Pythagoreans had not taught any such absurdity, but what was
correct. Thus Simplicius, in explication of the Aristotelian account
of the Pythagorean system of the world, writes as follows (and quite
similar words may be found in Asclepius and in an anon ;
scholium to Aristotle) :” oT
= is the way he understood the Pythagoreans’ theory himself.
But those of them with more genuine knowledge understand b
central fire’ the creative force which, from its mid positi ᾿
produces life over the whole earth, and keeps warm the εἶς of it
that tend to cool off.... And they used to call the ἐν τ sear”
because it too, like the stars, is a creator of time: for it is the cau
of days and nights. The part of it which is shone upon by the ᾿
makes day, and that which is in the cone produced b ὯΙ ἅν
makes night. The Pythagoreans called the moon Sse a
as though to call it “‘the ethereal earth,” and because it inter ἢ
the sun’s light, which belongs characteristically to the earth. ae
The expression about “more genuine” Pythagoreans here has been
cagcrly scized upon, because it seemed reasonable that there should have
been a geocentric system as a precursor to the more complicated system
described by Aristotle. Some have even asserted, mistakenl ΄ that
Simplicius is citing Aristotle,”? though the contrast is clearly pe
4 Aét. -nf,,d,, ΟΝ : ‘ a
and | heat eas ἄλλοι ones τὴν γῆν, In contrast to Philolaus, Heraclides, Ecphantus
‘ ; Pythagoras 1s obviously included among “the others.” :
Actius (2.12.1, 3.14.1; cf. Ὁ ong τῆς others.” If, according to
δ (2.12.1, 3.14.1; cf. below, ch. IV 1), Pythagoras divid
ee τον es heaven and earth similar]
er ς ZONES, os too presupposes a geocentric system. Aét. 2.29.4 ascribes the count :
car : ᾿ the Pythagoreans,” citing Aristotle (above, ch. 1 3, n. 26) =
iy, on Bae 138f, 141.128. On this Alexander, see H. Dahlmann and W. Spever
fee a Ie ie 2 Abh. Mainz 1959.11, 42; Burkert Philoloolis 1903 . opeyer,
O70, p. 215.7 Thesleff. Proclus cite is j πὸ ee es ἊΣ
198.348). s cites this in his interpretation of the Vimaenus (ΠῚ
Τ Vor refutation of the ide
‘ ἃ that the οἰ τῷ en a χεῦις,
urea earth moves, see Arist. Cael. 2.14, and esp. Ptol.
Simpl. Cael. $12.gff
ene ἀπ σώ DK §8b37; more brief iC 1 Ge το
4040 42 Wrandis. Cf ch. ΓΝ oo, ese ΠΟΥ ΝΟ Conk Ρ'
Th Pe OM: . . - τις
eo cae 247: Cornford, ‘Tim. 1a8; Kranz, RAM 1989, 141. Contra
ss, Plato s62, who suspects Tamblichus iv the source (referring, ec Mel Cal
a τς ᾿ ᾿ Sot source (referring to § ‘ae
$07. 12tf). Zeller judged the matter correctly, | g20f. ead
1. Philolaus’ Book in Pythagorean Tradition
between the “more genuine” Pythagoreans and Aristotle “himself.”
Also, there is no question here of the earth’s rotating.” The “more
genuine Pythagoreans”’ are represented as explaining three expressions:
ἐπὶ τοῦ μέσου πῦρ, the earth as ἄστρον, and the ἀντίχθων. These are the
distinctive ideas in Aristotle’s exposition of the Pythagorean system,
where they appear in precisely this order;”° and the counter claim being
made is that thesé words do not mean what they suggest to the un-
prejudiced reader. The “‘central fire,” they suggest, is a figurative
expression for the life-giving force which, emanating from the center,
permeates the universe; the earth is called a “star” because it is the
instrument of time—a simple quotation from the Timaeus’’—because
the shadow of the earth makes night.” Finally, the moon is called
“counter-earth” as being αἰθερία y#—an expression rather frequent
in the late tradition, bound up with the notion of astral immortality
and the moon-Hades.’8 In a word, the intent of the “more genuine’
Pythagoreans is to show that Aristotle basically misunderstood
Pythagorean astronomy, that it did not include a moving earth, and
that it did not imply anything other than the “normal,” geocentric
cosmology which had been dominant since the Hellenistic age. Their
method is a thoroughgoing allegorical interpretation of Aristotle's
words. Just as Aristotle’s exposition of the Pythagorean number
theory was countered with the “original” words of Theano,”® so here
the attack on Aristotle is armed with the claim of “more genuine”
doctrine.
In this way, then, an effort is made to expunge a Pythagorean
doctrine of terrestrial movement which was felt as an embarrassment. °°
74 Schol. Coisl. sosa3f, on the earth: τοῦτο δὲ τὸ ἄστρον φερόμενον νύκτα καὶ τὴν
ἡμέραν ποιεῖν... was taken as evidence of a moving-earth theory by Boeckh, KosmSyst
96; Duhem 89f; van der Waerden Astr. 58f (Heath, Aristarchus 250, thinks the scholium
is based on a misunderstanding). The basis is merely Arist. Cael. 293a22f, reinterpreted.
Τὸ Cael. 293a21, 22, 24.
78 Tim. 38c, 40c, 42d. Plutarch declares (Quaest. Plat. 1006e) that the passage does not
imply any movement of the earth.
77 Cf. Emp. fr. 48 (though he is not consistent; cf. Ajo), and the μαθηματικοί in Por-
phyry (Stob. 1.49.61; 1.€. professional astronomers; above, ch. I 2, n. 76).
78 Cf. Cumont, Symb. 187 n. 6. The moon as αἰθερία γῆ: Por. ap. Procl. In Tim. 1147
(presented as a teaching of the Egyptians; at II 48.17 further systematized); Macrob.
Somn. Sc. 1.11.7, 1.19.10 (a doctrine of “ physici”). ᾿οΟλυμπία γῆ: Plut. De def. or. 13.416e.
οὐρανός as ἀντίχθων: Clem. Al. Strom. 5.130. Ps.-Arist. fr. 245 (from a collection of
problemata whose origin is doubtful) gives a different interpretation of the doctrine that
the earth is a star and the moon is an earth, namely that moon and earth consist of the
same clements.—Only a part of the moon is ἀντίχθων, according to Plut. De fac. 29.944¢;
in Cic. Trsc. 1.68, the inhabited southern zone of the earth is ἀντίχθων.
7 Above, ch. [ 3, n. 52.
% Proclus sharply criticizes Aristotle for understanding the word ἰλλομένην in the
Timaeus as denoting a rotation of the earth (a Tin, ΠΠῚ 37.71).
1. PHILOLAUS
Aétius, however, distinguishes Philolaus from the “others” who did
not accept any movement of the earth—and he includes Pythagoras
here—and the astronomical system he ascribes to Philolaus corresponds
exactly to that which Aristotle ascribed to the Pythagoreans.®! People
have always been surprised at this remarkable fact, not noticing that
the situation is not unique. Not only in astronomy, but in relation to
the first principles and the derivation of the levels of being, Philolaus
stands in the doxographical tradition as the representative of the Pythag-
oreans described by Aristotle, though generally Platonic interpretation
has taken the place of what was really Pythagorean. Thus too, the
doxographical report that Philolaus spoke of the destruction of the
cosmos (φθορὰ τοῦ κόσμου) corresponds to Aristotle’s statement that the
Pythagoreans “without a doubt’ taught that the universe had a
beginning, whereas the later tradition projects onto Pythagoras the
Platonic-Aristotelian eternity of the world.® If it is necessary, on
historical principles, to abandon the Platonic interpretation of Pythag-
oreanism and realize that only the reports of Aristotle give reliable
indications of the genuine ancient teachings, it follows that in the
doxographical tradition the Philolaus items are all that remains
of what is genuine. And, since these very probably go back to
Theophrastus (we recall that Menon, another of Aristotle’s pupils,
cited Philolaus), there must have been a book of Philolaus available
in the fourth century that contained pre-Platonic Pythagorean
doctrine.
The agreement of the Philolaus tradition with Aristotle’s accounts
of Pythagorean doctrine has attracted the attention of many.® It can
be explained in either of two ways: either Aristotle gets his information
about Pythagorcan doctrine at least partly from Philolaus’ book,
SY AGt. 2.7.7, 3.11.3, 3-13.2 (= DK 44A16, 17, 21); cf. below, ch. IV 3.
δ2 Philolaus A 18 = Aét. 2.5.3. This disagrees sharply with ps.-Philolaus fr. 21, but
Stobacus (1.20,1-2) cites the two one after the other. On Arist. Met. 1091413, above
ch. I 3, n. 119; on Pythagoras ap. Aét. 2.4.1, above, ch. I 3, n. 117; cf. τάξιν ἄλυτον,
p. 165.4 Thesleff.
"9 Cf. Zeller I 369 ἡ. 3, ΚΙ, Schr. I 140ff; Frank 254, 290f, 318 n. 1, 327f; Bollinger 7off;
Mondolfo in ΖΜ 367ff; Raven, PyEl o8ff, KR 300ff. To collect the key points (cf.
ch. 12): πέρας-ἄπειρον, Philolaus Ag, Bt, 2, 6; even and odd, Philolaus Bs; ἀρτιοπέριττον,
Philolaus Bs; origin of the ἐν and of the world: Philolaus Bu, 6, 7; only one world:
Arist. fr. 201, Philolaus B17; the role of the number το; Arist. Met. 9868, Philolaus
A13, Theo Sm. 106.10; the astronomical system, ch. IV 3; the role of mathematical
music theory: Arist. Met. o8sb31, Philolaus Bo, A26, below, ch. V 2. On the connection
of the “breathing world” of Arist. Met, rograr3f, Phys, 213baa, fr. 201, with Philolaus
A27, above, ch. 1a, mn. 47.
1. Philolaus’ Book in Pythagorean Tradition
or the book was forged on the basis of Aristotle's accounts.** In -
first case the book must be regarded as genuine, that is, sg τὰ by the
Pythagorean Philolaus about 400 B.C. or a little earlier. ΤΣ is ue
basis on which to argue that the book Aristotle had was a orgery, ᾿
or that an authentic and ancient Pythagorean book had been eee
to Philolaus.8* Naturally, the question would still remain, whether a
the fragments we have come from this original work, or preier some
or all may come from a revised edition, an imitation, or even qlee
made to supply its loss. Nevertheless, the Philolaus tradition, πο
troubled, would even in this situation bear witness to a source ὁ
87
tremendous importance. |
Aristotle never names Philolaus, except when he cites that memorable
apophthegm in the Eudemian Ethics,®® and critics — made i" nae
principal argument against the authenticity of the iragments. Poe
totle used this book, they think, his silence is quite " mconcetva ᾿ ,
There is a prior question to answer, however: what were Aristot τ
sources for Pythagorean teachings? Original Pythagorean writings:
Oral tradition? Writings of the Platonists? .
This last possibility®° is excluded for most of Aristotle 3 τἀ κι
for, quite unlike the Platonists, he makes a sharp eareie ahaa
Pythagoreanism and Platonism. The role of oral tradition oug —
be underestimated, though here too Platonists were the intermediaries
64 So Bywater (50), following Schaarschmidt (15), and recently esp. Raven, PyEl
ff. Ch. III, 2, ἢ. 91.
ei ere 290f, 327f, and passim. He tries to show that the forger was Speusippus
‘olling: h. I 2.
- similarly Bollinger 44f. Against this, above, ς Ψ
Scat Ree bee II 93, thinks of an original Pythagorean ae later =
Aristoxenus which was then later fathered on Philolaus; but this neglects the close conn
i ith the account of Aristotle. eo ΝΞ ᾿ bas
ar The Philolaus fragments are sometimes cited in this spirit, as a “forgery which
nevertheless includes valuable material; as Ross, PTI 160f.
88 = DK 44B16; above, ch. If 4, n. 139. = 7
89 Τῆς cae “inconceivable” is Burnet’s (EGP 284 n. ἊΝ — jp eariiaiss
: hmidt 14, Tannery ¢ : :
Raven in KR 310, cf. PyEl 100; Schaarsc , Poly nig).
** Ari i kannten es [Philolaus’ boo :
2 Ι 88, Aristoteles und Aristoxenos kannten ¢s
atl aaa n. 140 (‘‘Aristotle’s silence certainly implies that he aes ae ee ne
book of Philolaus”), 386f. For a different answer, 566 Zeller (KISchr I 136-144 .
196 —-128. _
iii a poe at thought that Heraclides was a principal source ae
Frank ee Speusippus, along with other Platonists (258, 290f, 3276); a πὸ Υ
Howall and Bollinger. Speusippus may be the source for the “table of opposites "(a ;
: ᾿ . . + h
eee Archytas told about Eurytus may have sa ae a " εν: " Ἂς
ip: ~ ch. 1, n. 69). On the ἐὺ
Academy and the Peripatos (above, ς 2, ει
ciel in connection with Speusippus (Th. ar. 82.12), sce below, ch. |
II, PHILOLAUS
through whom he knew Pythagoreans, so that he is hardly likely to have
derived from this source the decisive points that he was able to make
against the Platonists. He expresses himself in very definite terms:
of μὲν οὖν “Πυθαγόρειοι πότερον οὐ ποιοῦσιν ἢ ποιοῦσι γένεσιν οὐδὲν Set
οισταζεῷ: φανερῶς γὰρ λέγουσιν ws.... Aristotle is citing τῆς
clear words”’ of the Pythagoreans in the discussion as to the sense in
which one may speak of a genesis of number and thus of the world as
a whole; and his polemic is aimed against a Platonizing reinterpreta-
tion.” The fact that he can insist on the exact wording in this wa
shows that he has a written source, for oral reports are not amenable :
such exactitude. The use of written sources is also suggested by the way
in which Aristotle cites Pythagorean technical terms, distinguishes
their statements from their “assumptions,”’ and sometimes even states
that this or that question remains unanswered.®? We cannot suppose
however, that the book or books he read claimed to be written be
“so-called Pythagoreans” (καλούμενοι Πυθαγόρειοι). The name of an
individual must have been attached to each. To put it bluntly, even
if we disregard the problem of Philolaus, the “inconceivable” is just
what happened. Aristotle based his account of Pythagoreanism on at
least one book, whose author he never named.
The Anonymus is not Archytas;® what Aristotle cites from him
indicates an advanced degree of abstraction, though the evidence is far
from plentiful. In addition to his books on the Pythagoreans, Aristotle
wrote three περὶ τῆς ᾿Αρχυτείου φιλοσοφίας, and this may be taken to
show that he distinguished clearly between Archytas and general
* Met. 1og1a13ff, above, ch. I 2, ἢ. 41. Zeller pointed out the significance of the
a ieee ). Schaarschmidt (82ff) denied that Aristotle had written sources
» χροιά, Serts. 439230 (above, ch. I 3, n. 96), μίμησις, Met. 987b11 (above ch.I2,n 82)
ἐξ ον γὰρ ὑποτίθενται καὶ λέγουσιν ..., Met. 990a14, .. . ἀπορεῖν ἐοίκασιν, Met To8eba0:
| Pace Frank 361: ‘‘Archytas seems to have been the only Pythiaporesn from whose
writings Aristotle learned about Pythagorean philosophy.” This is repeated by Wuil-
leumicr, 577. Cf. also Frank 135f, with n. 387, and pp. 77, 196. Aristotle’s nee of
Archytas, though, do not yield much: Pol. 1340b26 (DK 47At10) ᾿Αρχύτου πλαταγή
ἃ childs toy (Frank, 339, rightly compares Pl. Leg. 653d); Rhet. en. (DK AWD
ταὐτὸν εἶναι διαιτητὴν καὶ βωμόν, an apophthegm that is reminiscent of the acusmata
The “definitions” of Archytas (above, ch. I 2, n. 104) take us a bit further. At Arist fr.
199 -᾿ Theo Sin. 22.5ff, συμφέρεται δὲ τούτοις καὶ ᾿Αρχύτας (DK 47A21), is Gbyiously
an addition by Theo (who on another occasion connects Archytas with Philolaus: bow,
n. 64). Frank states this, though with hesitation (257 n. 2), only to affirm later (361)
that Aristotle “cites as the special doctrine of Archytas (fr. 199 Rose) a view which in
another passage (Met, gg6ar8) he presents as general Pythagorean.”—Philosophically
interesting are Archyta:
fr. 65). : chytas A23 (ch. I, 2, 0. 106), Azza (I 3, 1. 156), and Azq (Eudemus
1. Philolaus’ Book in Pythagorean Tradition
Pythagoreanism.* Archytas on occasion mentioned earlier Pythag-
oreans,% but he was an independent scholar and surely was not
concerned to write as detailed a doxography as would have been
necessary if Aristotle were to get all his facts about Pythagorean
philosophy from it.
The only name that is closely connected, in Aristotle's reports,
with the philosophy of the “so-called Pythagoreans,” is that of
Eurytus; but he seems himself to have essayed an extension of the
number theory. Now, from Aristoxenus on down, the tradition
constantly associates Philolaus and Eurytus.®” Philolaus was the teacher
of the Pythagoreans from Thebes and Phlius whom Plato knew. So we
see the skimpy biographical evidence on Philolaus converging with
what can be deduced from the doxography about his book; namely,
that it was to Aristotle, in a. way, an authoritative exposition of
Pythagorean number theory.
Philolaus does not appear as a rebel against the traditional obligation
which Pythagoreans and, to a certain extent, Platonists felt, to re-
cognize that all doctrine originating in their own minds was only
extension and confirmation of the ancient wisdom of Pythagoras.°° At
least his pupils, and Plato too, must have understood Philolaus’ work
in this way. It is only from this point of view that we can understand
Aristotle’s strange uncertainty about the chronology of the Pythago-
reans, as well as his silence about the originator of the system. Sub-
sequently the Platonic view of Pythagoras became dominant, and that
book was no longer an accurate presentation of Pythagoreanism, as it
was now understood. Its doctrines now appeared to be the private δόξα
of the author, Philolaus.
The thesis, on the other hand, that the book of Philolaus was a
forgery that took its materials from Aristotle takes no account of the
general development of the Pythagorean tradition. It is simply not
true that the picture of Pythagoreanism standard in late antiquity was
derived in its essentials from Aristotle’s exposition.°® The Platonic
96 Cf P. Moraux, Les listes anciennes des ouvrages d’ Aristote (Louvain, 1951) 106, 201,
301 (DK 47A13). Frank (n. 335) considers referring to this book Philop. Aet. mundi
p. 522.20 (on the πέμπτον σῶμα). Some kind of connection between Archytas and the
‘Timaeus is indicated by the title "Ex τοῦ Τιμαίου καὶ τῶν ᾿Αρχυτείων (no. 94 in D.L.
5.25, no. ὃς in the anonymous catalogue); cf. above, ch. I 4, nn. 4-5, I 3, n. 166.
% CF fr. 1 and the mention of Eurytus (above ch. I 2, n. 69).
8? Above, ἢ, 50.
98 Above, ch. 1 4, n. 26.
%” The view of Schaarschinidt, p. 15.
Hil, PHILOLAUS
interpretation is dominant, and the evidence of Aristotle was ignored
distorted, or rejected.1°° The “‘forgeries” were not intended to deceive
philologers, but to provide a cachet of age-old wisdom for a doctrine
adapted to the needs of the writers’ own times. This is the reason why
the pseudo-Pythagorean writings take Platonism as their basis. If the
Philolaus reports alone, along with some material from Archytas
form an exception to this general trend, then their agreement with
Aristotle is not a reason to suspect them, but quite the opposite: an
indirect proof of their authenticity. Of course, only a keen and attentive
study of all the details, especially in the directly quoted fragments, can
enable us to decide, in each case, what reason there may be to ἜΠΗ
that they are authentic or spurious. But the external tradition itself
shows that the situation of Philolaus is different from that of the great
mass of would-be Pythagorean authors. A book made him famous:
it was still in existence in the fourth century; and it showed traces at
least, of ancient, pre-Platonic Pythagoreanism. |
2. THE SPURIOUS AND THE GENUINE IN THE PHILOLAUS FRAGMENTS
If any genuine fragments of Philolaus’ book have been preserved
they are the most important, because the only original, documents of
carly Pythagorean philosophy. This statement can easily lead to a double
misunderstanding. One expects to find bedrock, so to speak, forgetting
that Philolaus was a contemporary of Socrates and Democritus
Gorgias and Diogenes of Apollonia, and that he wrote and taught ὅν
only after Parmenides, but later than Zeno, Melissus, Anaxagoras
Empedocles, and Protagoras.! Again, one is likely to expect to aud
thoughts of striking and unique originality,? overlooking the fact
that in the second half of the fifth century, thinkers were much more
concerned with assessing the many original but contradictory assever-
ἠῷ It was modern critical method, especially the work of Zeller, that first set Aristotle’s
gee clearly distinguished from the Platonizing and neo-Pythagorean conception
ae eae a Lp of authority which, as is easy to forget, it certainly did not
’ Diels, Hermes 1893, 417ff, even considered the etymology φλέγμα-φλέγειν in Menon’s
excerpt (A27-28) as borrowed from Prodicus (fr. 4); cf., per contra, Fredrich 37 n. 1
Olivieri 20f, 45f—Frank (304) states it is impossible to suppose that Plato harked back.
in the Philebus, to a book “almost too years old’’; but at the time of the compositi
of the Philebus Philolaus’ book was scarcely more than 50 years old. μὴ
* Bywater (29) does not see in the fragments an “ original effort of mind,” and athetizes
them on the basis of this “test of excellence.” But maybe what was original in Pytha
oreanism was not philosophical expression, | mn
2. Spurious and Genuine in the Philolaus Fragments
ations of their predecessors—comparing, reconciling, adapting. It was
no longer a day of lonely prophets but one of far-reaching debate.
A statesman like Melissus of Samos might take part in it, or a poet like
Ion of Chios; physicians, too, were beginning to formulate in written
terms the scientific basis of their art. A book by a Pythagorean, in this
period, cannot have been so much like an erratic boulder asa link in a
long chain of tradition.
We are inclined to see in Philolaus nothing but the transmitter of
older Pythagorean teachings, and in fact all indications are that the
Pythagoreans themselves saw nothing more in his book than an
expression of the wisdom of Pythagoras. But before Philolaus there
was no written exposition of Pythagoreanism;? and the transition from
oral to written teaching is much more than a matter of externals; it
means a fundamental transformation, of content as well as of form.
An orally transmitted doctrine, in spite of all the special training of the
memory in ancient times, notably among the Pythagoreans, must
always be enmeshed in the very fabric of life, always exposed to
psychological forces that can mold and transform it. No matter how
persistently one holds to a traditional knowledge, even if the basic
attitude and intentions remain unaltered, changes of detail will appear.
As needs change, from time to time, different aspects will move into
prominence. Oral teaching is always directed toward specific learners,
and therefore its nature is determined by the listener as well as the
speaker. In particular, the form of oral transmission will alter, because it
must each time be impressive—except that poetic form guarantees a
certain consistency. On the other hand, a written work is free from the
restraint of a specific situation of speaking and hearing. It makes a
claim to be valid in itself, independent of any special καιρός, and not
needing any further intervention by the writer. A book in prose,
furthermore, renounces the embellishment of verse, because its
message is supposed to be dependent entirely on the subject matter,
and not at all on any predetermined formal structure. The first prose
book to prefigure and exemplify this development was that of Anaxi-
mander.‘ It is only when linguistic expression attains this kind of
objectivity that scientific or philosophical discourse really becomes
possible; from the time of Anaximander, Greek science and philosophy
% Above, ch. IE τ.
41K 12A7; cf. At. Pherecydes of Syros, who is also named as the first writer of prose
(Sudas.v. DK 7A2), may be dependent on Anaximander (von Britz, RE XUX 2ojof).
eest CS SOF OE Ete Be!
ra pl pe ele ese he τως
te | arly recognized—that, if Pythagorean
doctrine was not committed to writing before Philolaus, then there
did not exist, before Philolaus, any Pythagorean philosophy, in the
Greek sense of the word, but only a different kind of thing: a lore or
wisdom” consisting of disconnected teachings about the world
gods, and human beings, having its foundation in a specific wa of
life and transmitted in individual maxims. This “wisdom” and ie of
life were variable in detail and lacked logical foundation or Seen
and conceptual coherence; in fact they consisted in our familiar
acusmata, the doctrine of transmigration, and the βίος Πυθαγό
in which they were rooted. μὰ
If a Pythagorean, in a situation like this, undertook to write περὶ
φύσεως, this was ἃ μετάβασις εἰς ἄλλο γένος. It meant the adoption of a
manner of exposition foreign to the Pythagorean tradition, the product
of a different kind of development. The tools for such eae εἰς
concepts, definitions, modes of argument—come from a non-Pythag-
orean background. What we should expect to find then, would ᾿
be ἃ unitary and original product, but a hybrid eclecticism,® a con-
glomeration of Pythagorean attitudes and borrowed conceptualization
For there is clearly nothing in the Pythagorean teaching itself that oui
demand written consolidation, or formulation in philosophical and
scientific terms.
August Boeckh decided, with relation to the Philolaus testimonia
as a group, that “the only solution is to recognize all we have as
genuine or to reject all of it as spurious.””? Those who doubt its authen-
ticity have been happy to cite this sentence, for it opens up the possi-
bility of affecting the whole by an attack ona part—if they could ae
one column, the whole structure would collapse.* But this is a hasty
᾿ "τῇ
ὙΠ Zeno visited Athens, he brought his γράμματα with him, and people gathered
as im to get to know his book (Pl. Parm. 127¢; the report is not likely to be historic-
: ΤΕ but eae what must have been a possible occurrence). In his depreciation
of the written word, ili i
κει ato is rebelling against the prevailing fashion, in favor of older
; ᾿ ;
᾿ oe oe. ps.-Arist. MXG 979a14: συνθεὶς τὰ ἑτέροις εἰρημένα. C. Schick
7 rch. g ottol. ital. 40 (1955) 128ff, thinks that the book bearing Philolaus’ name <
eee a tradizione interna della scuola pitagorica,” a sort of Doric jargon belon =
ὦ t ; ΡΝ" τς is not proven by Por. VP 53 and Iam. VP 24 τί, any more Hak by
᾿ς alleged echoes of it in Parmenides and Z
ee eno (below, ch. III 3).
: a
Seiden 2f, 79 (citing Boeckh), Bywater 50, Frank 290. No one any longer
thinks of applying the same procedure in the case of Archytas, where the spuricus need
amounts to more than 10 times as much i ; -
Il τ n. 14). as the little bit that is genuine (see above, ch.
240
oversimplitication, especially since Boeckh’s conciliatory assumption
that Philolaus wrote a single work in three books has turned out to be
based on a misunderstanding. Hermippus speaks of “one book” of
Philolaus,® and Demetrius of Magnesia gives its title, Περὶ φύσεως, and
cites the opening line, an established bibliographical custom.° This
citation is closely related to the long excerpt in Stobaeus, to which he
gives the heading Περὶ κόσμου, but from which Nicomachus gives a
fragment with the label Φιλόλαος . . . ἐν τῷ πρώτῳ φυσικῷ." Stobaeus’
evidence for the title is not important, for his own section, within
which Philolaus is cited, bears the heading Περὶ κόσμου. Nicomachus
in the Theologumena has a further fragment of a Περὶ ddcews,” and this
title is also mentioned by Theo Smyrnaeus.”* Proclus!4 and Boethius’®
are both doubtless dependent on Nicomachus. The doxographical
9 Above, ch. [I 1.
10 D.L. 8.85 = DK 44B1: τοῦτόν φησι Δημήτριος ἐν ‘Opwvipors πρῶτον ἐκδοῦναι
τῶν Πυθαγορικῶν «βιβλία καὶ ἐπιγράψαι» Περὶ φύσεως, ὧν ἀρχὴ ἧδε. Diels’s supplement
can scarcely be right, and his reference to D.L. 3.9 is certainly a mistake (above, ch. III 1,
n. 28). Perhaps there is no lacuna, and the object of ἐκδοῦναι is the title (cf. Strabo 1,
p. 15: ἡ περὶ τῶν ἀγαθῶν ἐκδοθεῖσα ὑπ᾽ αὐτοῦ πραγματεία; Plut Rom. 8 [ὁ Πεπαρήθιος
Διοκλῆς] ὃς δοκεῖ πρῶτος ἐκδοῦναι “Ῥώμης κτίσιν). The indefinite plural ὧν does not
necessarily mean that more than one book is intended. After all, only one “beginning”
is being quoted.—In a similar way, D.L. cites the incipit of books of Diogenes of Apollonia
(9.57, 6.81), Pherecydes (1.119), Alcmaecon (8.83). At bottom lies the practice of the
library at Alexandria (U. von Wilamowitz-Moellendorff, Antigonos von Karystos [Berlin,
1881] 323; Wiersma, Mnemosyne 1942, 23).
11 Stob. 1.21.7 = DK 4482, 4-7. There was another excerpt in the lost prologue of
Stobaeus; the only trace remaining is the marginal note, Φιλολάου (Stob. I p. 15.4 App-).
Nicom. Ench. 9, p. 252.13ff (the title cited); pp. 252.17-253.3 = DK I 409.10-410.3.
Evidently the short, very corrupt citation in the Hagiopolites is from Nicomachus (M.
A. J. H. Vincent, Notices et extraits des manuscripts XVI 2 [Paris, 1847] 268 = DK 44Bé6,
Ι 409.10f: ὁ Πυθαγορικὸς Φιλόλαος, ἔν τινι πονήματι αὐτοῦ, πρός τινα γυναῖκα Πυθαγορείαν
ἐκτιθέμενος. See also Tzetzes, cited ch. ΠῚ 1, n. 28.) Further, Nicom. Ench. 12, p. 264.3
(= fr. 6, DK 1 410.1f); Ar. 2.18.4 (= B2, DK I 407.2f) (translated by Boethius in Ar.
2.32, p. 126.7ff); Nicom. Ar. 2.26.2 (= Α24). See next note.
12 Th. ar. 25.17 (= fr. 13); on 74.10 (= A12), see below, nn. 41-44.
18 196.10 (DK ad fr. 11; below, n. 169).
14 Procl. In Tim. 1 84.4, 1 176.28ff (ἐκ περαινόντων καὶ ἀπείρων = frr. 1-2; cf. Nicom.
Ar. 2.18.4), Theol. Pl. 3.7, p. 132 Portus. Probably the expressions δίεσις and ἀποτομή
are from Nicomachus (Procl. In Tim. II 168.28f; the source is given as τῶν παλαιῶν
τινες); Il 190.7ff (cf. A26, B6); perhaps also the report about the gods of various angles
(A14; cf. below, ch. IV 3); cf. Theol. Pl. 1.4, p. 9 Portus: καὶ yap τοὺς ἀριθμοὺς
ἀνεῖσαν τοῖς θεοῖς καὶ τὰ σχήματα, καθάπερ λέγουσιν οἱ τὰ ἐκείνων ἱστορεῖν σπουδάζοντες .--
Damascius is probably drawing directly on Proclus at Princ. 1111.12 Ruelle (ἐκ περαινόντων
καὶ ἀπείρων), cf. 1 101.3: ἐν τοῖς περὶ φύσεως. Also, Damascius may have drawn his
information about the dedication of certain geometric figures to various gods (A14)
from an account of Proclus which has been lost.
15 Cf. above, n. 11. Boethius surely derived even details of music theory from Nico-
machus, whose musicological writings are only preserved in fragments (Mus. 3.5 = A26;
3.8 = n. on B6). Iamblichus, too, can have taken his Philolaus citation from Nicomachus
In Nic. 7.24 = fr. 3; repeated by Syrianus, at Met. 147.17).
241
Hl, PHILOLAUS
reports in Aétius’® cohere closely with these citations, suggesting the
probability that this book Περὶ φύσεως from which eacaa of
Magnesia (time of Cicero), Nicomachus, and Stobacus quoted was the
one that Theophrastus read.
nog must ᾿ the watchword, however. Nicomachus knew of
more than one book attributec *hi
“first.” Stobacus has a oleae ae ᾿ ᾿ ad ᾿
χῆς,11 and both
seen and ears mention a title Φιλολάου Βάκχαι.18 Claudianus
amertus speaks of “many books” that Phil
a “third book περὶ ῥυθμῶν = μέτρων.᾽᾽1.9 poe pap tes “ang
long passage given without title by Stobaeus,2° and various isolated
citations.
What the tradition offers us, then, under the name of Philolaus. is
not a unificd picture, but a great variety. The first task that su cats
itself is to demonstrate its homogeneity, but there seems ede aes
prospect for success in that, if only because of the open contradiction
between the “one book” of the older tradition and the plurality of
books in the later. It is true, though, that from the beginning a ΒΞ
nucleus seems to be discernible: fragments 1, 2, and 4-7, along with the
accounts of Aétius and Boethius, dealing with ontology, astronomy
and music theory. |
We may take as the starting point for a closer examination of the
tradition the fact that onc of the longer fragments is certainly spurious
namely the paragraph on the world soul (21). Since Zeller erenouned
it spurious, no one has seriously defended it, and it is unnecessary to
repeat all the arguments.” This passage is an example of Hellenistic
14 δε: i ΕΣ]
: ic 15-21. The “indirect” proof of authenticity applies for these accounts (above
ch. 1).—Also from the doxographical tradition are the notes in Cens. 18.8 (A22),
Macrob. Somm. Sc. 1.14.19 (A23), Sext. Emp. M
mae ᾿ . . th. 7. .
Stal 4b Se OL. ) p. Math. 7.92 (A29; below, n. $5).
Err. 17-19; below, nn. 140-148.
fr, 22; below, ἢ. 45.
Ὁ 1, 1-12; below, nn. 169-185.
4 Zeller 1 369.3, 476.1, KiSchr | 145ff; Schaarschmidt 24; Bywater 4off; Frank 282ff;
Moreau, Aime 145}, Thesleff, Texts 1sof. A few points may be em hiasized b he
with the Aristotelian doctrine of the eternity of the world ἐξ δ δος εἰ " "aad the
presence of technical terns like ἐνέργεια (IDK 1 418.6), τῷ γεννήσαντι mictnear ἌΝ
(Ι 418.1 Is cf. Pl. Tim. 37¢; here even Boeckh is forced to admit the ρου θρθο ὀ νὰ
terminology; cf. Frank 290 ἢ. 1), φύσει διαπνεόμενος (I 417.14) like the Stoi am
διῆκον (tor such Stoic coloration in later Pythagorcanism, see Cic. Nat d a se
Kinp. Math. 9.127). At 1 417.14, the meaningless ἐξ ἀρχιδίου of the MS ς ἡπρῖ ες “ἢ
be replaced by ἐξαρχίδιον (as in the inscription cited ad loc. in DK C IG 42 SIG
712.1). It is an adjective, formed after ἐξ ἀρχῆς as ἐξαιφνίδιος after ἐξαίφνης, and is
242
y. Spurious and Genuine in the Philolaus Fragments
“cosmic piety,” the resultant of the hymn-like eloquence of the Ti-
maeus and Aristotle’s doctrine of the eternity of the universe. Signifi-
cantly, the fragment has a good deal of wording like that of Ocellus.
If Harder is right that it is Ocellus who is derivative,” then this frag-
ment must have been composed, at the latest, in the second century B.C.,
and the one certainly spurious fragment would be the earliest attested.
The decisive question is whether this fragment is closely connected
with the rest. Frank tried to prove this,?* but his arguments do not
suffice. There are points of contact with some elements, contradictions
of others. In contrast to the hymn-like celebration here of the eternity
of the world, ἐξ αἰῶνος καὶ εἰς αἰῶνα, where the “creator,” as in the
Academic tradition, is to be understood as a didactic concept, not
a reality,24 one of the other doxographical notes speaks about the φθορὰ
τοῦ κόσμου,25 and expressions like ἁρμόχθη, συνέστα, ἤρξατο γίγνεσθαι,
and ἁρμονία ἐπεγένετο in the word-for-word citations*® presuppose an
origin for the “order” of our world. The contradiction is evident.
Under the rubric Περὶ τάξεως τοῦ κόσμου, Stobaeus®” first ascribes
to Philolaus the cosmic system known from Aristotle: the order is
See ge ι57τ-ΛῈ“σ.΄ὁ!,ὔδυ:ρ..-ς-
predicative with dpxdy in line 13. On the question of the ἀρχὴ κινήσεως see Arist. Cael.
285b5ff.—Macrobius cites as the doctrine of a secta of Platonici sentences that show exten-
sive verbal agreement with Philolaus fr. 21 (Somn. Sc. 1.11.5f; noted by Capelle, De
luna 9).—Frank, in accordance with his general theory, must credit this fragment to
Speusippus, although this compels him to consider the possibility of “‘spatere Retuschen”
(290). Moreau (Ame), attempting to find intermediate steps between Plato and the cos-
mology of the Stoics, would like to date the fragment in the age of Aristotle (149),
making it approximately contemporary with the De caelo; but he cannot find convincing
parallels to φύσει διαπνεόμενος (pp. 136ff).—Rostagni (Verbo 53) maintains that at least
in content the fragment preserves genuine material, in spite of its late style; one wonders
f it is not rather the case that Rostagni’s version of Pythagoreanism shows Platonic
features.
22 The parallels are collected by Schaarschmidt (20), Bywater (39f) Harder (958, 119f),
and Beutler, RE XVII 2364ff. Harder supported his argument for the priority of
“Philolaus” with a faulty combination (above, ch. HI 1, n. 34), leading Beutler (RE
XVII 2363) to deny it; but the smooth style and perspicuous organization of the
“Philolaus” fragment make its priority seem likely.
23 28>ff. Contra, Mondolfo in ΖΜ 376f. It is admitted that there are points of agreement
with the second part of Ατό (cf. below, τ. 37), A17 (below, n. 38), and fr. 22. But it is
wrong to interpret ἀρχὰ ξυναπάντων in fr. 13 as “world soul”; ψυχά (= “life”’) is explicitly
distinguished from it. The emphasis on the one world (fr. 17; presupposed in frr. 1, 2,
and 6) does not prove anything; this is an old controversy (Aét. 2.1.2-3), and Aristotle
ascribes the doctrine that there is but one world expressly to the Pythagoreans (fr. 201); cf.
below, n. 26.
24 Above, ch. I 3, n. 118.
25 Arg: cf. ch. ΠΠ 1, ἢ. 82.
26 Err. 1,6, 7, 17; cf. Arist. Met. 1ogiat3ff, below, ch. 13, n. 119.——Frank (288f) equates
the eternity of the world in fr. 21 with the eternity of being in fr. 6; cf below nn. 86-87.
27 Stob. 1.22.1d Aét. 2.7.7 (inissing in ps.-Plut.) DK 44Ar6.
243
1. PHILOLAUS
᾿ /
OUPAVvosy i : ἱ ‘tS, Si
paves, then the five planets, sin, moon, carth, counter-carth, and
ἑστία the central fi | ia
ὁ central fire. But then he goes on:
τὸ μὲν οὖν ἀνωτάτω μέρος τοῦ περιέχοντος, ἐν ᾧ τὴν εἰλικ ἰνειαν
etna may το χειῶν, "Ολυμπον καλεῖ, τὰ δὲ ὑπὸ oy τοῦ OMumov
iad ve τοὺς πέντε πλάνητας μεθ᾽ ἡλίου Kal σελήνης τετάχθαι
pene τὸ δ᾽ ὑπὸ τούτοις ὑποσέληνόν τε καὶ περίγειον μέρος, ἐν 3 ὰ
τῆς φιλομεταβόλον γενέσεως, οὐρανόν καὶ περὶ μὲν τὰ oe ae
a βετεώῤων γίνεσθαι τὴν σοφίαν, περὶ δὲ τῶν a
ἀταξίαν τὴν ἀρετήν, τελείαν μὲν ἐκείνην, ἀτελῆ δὲ ταύτην
This passage contradicts what precedes, first of all, in terminolo
What was οὐρανός there, has to be called "Ὄλυμπος, to make the
contrast with οὐρανός. But a textual correction will not solve th
difficulty.28 First we have the fiery envelope of the world (zepié ἢ
but then it becomes στοιχεῖα, in the plural. And if the sublunar ‘ veal 3
is to be called περίγειον, the point of view is obviously Rees
and the express designation of this area as clea lee ἐς
central fire out of account. Therefore we have here elements of t
mutually exclusive systems combined in a single passage. i
Attcmpts at compromise will not do;?® onc of the two parts must be
rejected.®° The designation of the heaven as "Ολυμπος3 and the separ
tion of the supralunar and sublunar realms are commonly thou he of
as genuincly carly Pythagorean,® and for this reason Wiersma decided
28 Diels, DK 1 403.1
.17, replaces the manuscript reading wi ὑρανό
! ἱ : g with [οὐρανό L τὴν τῶ
elo σφαῖραν», referring to Arist. fr. 203 = Alex. Met ᾿ I Regine per
ae ae (101) found the contradiction “unbegreiflich,” but tried to smooth over the
: ie Υ as did Zeller, I 548 n. 1. Immisch (72 n. 1) speculated on a “development”
μ᾿ res thought. Schaarschmidt (28ff) and Bywater (38f) thought no confusion w
EW this forger. Frank (279f) ignored the contradiction. =
Or, of course, both.
81» x
ἣν ΠΣ ἔσχατος, Parmenides fr. 11.2 (also γάλα οὐράνιον); ᾿Ολύμπιος κόσμο
bd. 2; |
a sabes ean aaa Νὰ 44 (cf. Soph. OC 1655). These three testimonia add
ὶ : robability that the expression was Pytha in origi
need be assumed as basis than Ce eee μον
ss ὲ an acusma like that about the Isles of
pia ere an e Isles of the Blest (above, ch.
: _ IV 4). It was natural to identify the mythi
thical Olym h th
sky, and this took place lon : : ἦν: eahel neater tin’
his ὲ g before Pythagoras: the θεοὶ ᾽Ολύ i
the οὐρανίωνες (Pl. Epin. 977b: cf. below ἃ 36) Hie νος δου ἐϑυων
HP] A Tae Ue
Pes aes = es Al2 and fr): Heraclitus spoke of the impurity of the moon’s
ie a : 9. ο, Αξε, 2.28.6, Hippol. Ref. 1.4.3 = Emp. A6z2). For the moon as μέση
: ( a ae juncture of the regions of the cosmos, see Hebd. 2. This doctrine i
regarded as carly Pythagorean by Kranz, f | ᾿
: ἢ or example (NGG 1938, 141). But the 1
twocosimic realms could only have ΟΥ̓͂Ν: ] τ eae
ν ἶ y have become widely influential after the scienti
Oe one after the scientific astronomy
Ἢ agorean acusma about the plancts as the ἢ]
pea σε εν an αἱ : ᾿ rounds of Persephone”
δὰ in Pale eee direction (below, ch. IV 1). The doctrine of two worlds ae
γραίαν after the work of the Old Acaden aoe
ademy and Aristotle; then it be s
ee eg adem ξ : xecomes ΠΟ ΠΟ, in
a Pythagorean pscudepigi apha, c.g. “Philolaus” fr. 21, Ocellus 37, Hyponi. 26, Ano
γος. 439b20f; see also Epiphanius De fide 9.12 (Dox. $90.11) ΝΝ "
244
the second part was genuine. But the stronger arguinents are on the
other side. The Pythagercan origin of the ideas of the second part 1s
only a conjecture; it is through Platonism that they had their influence.
On the other hand, the cosmology of the first part is guaranteed
Pythagorean by the testimony of Aristotle, while later Pythagoreans
tried to get rid of ἰτ.38 The ‘dea that the highest heaven consists of the
εἰλικρίνεια τῶν στοιχείων COMES from interpretation of the Timaeus.®®
‘The threefold division of the cosmos corresponds to ideas of the
Platonists,3® and the expression φιλομετάβολος γένεσις shows in both style
and content a relationship to the fragment on the world soul.*” Thus
the sccond part bears the mark of Platonism, while the first corresponds
to the Pythagorean doctrines attested by Aristotle, so that there is no
question of the second part being authentic; this question rises only in
the case of the first. Finally, it 1s likely a priori that if one part is a
33 Mnemosyne 1942, 25. The first part is explained as the product of an error by
Theophrastus, who, he thinks, worked the later system into his interpretation of
Philolaus.
34 Above, ch. III I.
35 Tim. 32bff, Por. ap. Philop. Aet. mundi 13.15 and Cyril. Adv. Iul. 2.47e, Procl. In
Tim Il 43.20ff: πάντων ai ἀκρότητες make up the substance of the οὐρανός (If 49.15;
εἰλικρινὲς πῦρ ἐν οὐρανῷ, Il 44-1), schol. Pl. Phd. 109b p. 235 Hermann (from Ruhnken;
missing in Greene’s ed.): λέγει ἐν τῷ Τιμαίῳ τὸν οὐρανὸν τῶν τεσσάρων εἶναι στοιχείων,
ἀφ᾽ ἑκάστου τοῦ καθαρωτάτου .. -
86 The Timaeus separates terrestrial γένεσις from the regularity of the world of the
stars just as it does the ποικιλία of the planets from the ταὐτοῦ φορά of the fixed stars.
The resulting tripartition was emphasized by the ancient commentators (Adrastus ap.
Theo Sm. 148.13ff; Anon. Phot. 439b17ff also gives a special status to the ἀπλανές
superior to the planets; cf. Immisch 72ff). Xenocrates, somewhat differently, brought the
threefold division of the world (sublunary region, region of stars, or heaven, and highest
heaven, or what is beyond the heaven) into connection with divine powers and psychic
functions (ἅτ. 5, 15, 18 H., and “what the Delphians say,” Plut. Quaest. conv. 7458;
cf. Heinze 758). Heraclides (fr. 95) divides the cosmos into the realm of Zeus, that of
Poseidon (the spheres of the planets down to the sun), and that of Hades (from the moon
on down). Arcesilaus distinguishes three classes of gods, “ Olympios, Astra, Titanios,”’
which last group probably corresponds to the ὑποσέληνοι δαίμονες in Xenocrates fr. 15
(Tert. Ad nat. 2.2; Varro, following Antiochus, ap. Aug. De civ. D. 7.6). Pl. Epin. 977b
clearly uses κόσμος, Ὄλυμπος, and οὐρανός as synonymous. At 986c the word κόσμος
connected in an emphatic way with the paths of the stars (cf. Arist. Met. 1063a1 4), and
at 987b the heaven of the fixed stars is said to be κόσμος in the truest sense (cf. Ach. Is.
p. 36.5, where the Timaeus is referred to). At 976d the οὐρανός is celebrated as the provider
of time and therefore of true ἐπιστήμη; it is contrasted with imperfect ἀρετή, which is
possible even without λόγος (977 c-d). Here are the elements for the theory attributed to
Philolaus; it was natural to combine these points with the threefold division of the world
and the trinity of its appellations.—One also could read into a verse of Homer the idea
that "Ὄλυμπος was above the οὐρανός (Il. 1.497; cf. Stob. 1.22.2, ps.-Plut. V. Hom. 95;
Delatte, Litt. 126).
87 On this point even Boeckh admits post-Platonic. terminology (100). Cf. fr. 21 (1
418.3), where the words γενέσιος καὶ μεταβολᾶς are used to characterize the sublunary
sphere.
245
Hl PHILOLAUS
later addition, corresponding to later cosmological views, it should
follow the other.®8
Thus there are, in the Philolaus fragments, at least two different
strata, onc that has its origin in the Hellenistic apocrypha and another,
sometimes contradictory, which agrees with the reports of Aristotle.
Whether and to what extent these fragments contain original material
from the fifth century B.c. is a problem we can only approach by
careful study of the directly quoted fragments. But, as a preliminary,
let us eliminate whatever has nothing to do with this central question,
Speusippus’ book Περὶ Πυθαγορικῶν ἀριθμῶν contained thoughts of
Speusippus which, according to the testimony of Aristotle, are not
Pythagorean.*® The quotation from the book is introduced by a
detailed account of his sources: Lmevourmos...ék τῶν ἐξαιρέτως
σπουδασθεισῶν det Πυθαγορικῶν ἀκροάσεων, μάλιστα δὲ τῶν Φιλολάου
συγγραμμάτων, βιβλίδιόν τι συντάξας γλαφυρόν. .. . It is unlikely, con-
sidering the way it is phrased, that this reference to Philolaus comes
from Speusippus himself.4° A reference of this type would be almost
unique in ancient literature.
38 The citation of Ocellus at Aét. 2.25.13 is probably an addition of Stobaeus. Cf.
Diels, Dox. 100.1, Harder 39. Also in A16 = Aét. 2.7.7, Stobaeus may have made
additions on his own. (The parallel citation of the ps.-Plutarch is missing here.)—The
second part of A16 was rejected by Heinze (74 n.1) and Doring (AGP 1892, 517£).—A17
(Aét 2.4.15) presents problems similar to those brought by Ατό. In the sentence τὸ δὲ
ἡγεμονικὸν ἐν τῷ μεσαιτάτῳ πυρί, ὅπερ τρόπεως δίκην προὔπεβάλετο τῆς τοῦ παντὸς
σφαίρας ὁ δημιουργὸς θεός, the word ἡγεμονικόν may have been inserted by a doxographer
to correspond to the rubric ποῦ ἔχει τὸ ἡγεμονικόν (ὁ κόσμος); But δημιουργός too,
used in this sense, is impossible before Plato (C. M. A. van den Oudenrijn, Demtourgos,
Diss. Utrecht, 1951; C. J. Classen, C&M 23 [1962] 1-22). Is this another case of a later
term creeping in (Boeckh 96f, Wilamowitz, Platon II 89)? Or has there been a reinter-
pretation to introduce conformity with the predilections of the “ more genuine” Pythag-
oreans (above, ch. III 1, ἢ. 72) ? The sentence follows Ar8 in Stobaeus. On the simile of
the laying of a keel, cf. Pl. Leg. 803a-b.
39 Speusippus fr. 4 = Th. ar. 82.10ff = DK 44A13; above, ch. I 3. It is hard to decide
whether Iamblichus had the Speusippus passage from Nicomachus (cf. above, ch. If 1,
ἢ. 4), or whether he actually had Speusippus’ book. He used Speusippus in writing Comm.
math. sc. (Merlan, PiNeopl 96-128).
40 Frank (140f, 310, 332) and Cherniss (Pres. 390) take it as self-evident that Speusippus
referred to Philolaus; this is denied by Wilamowitz (Platon II 88) and Theiler (Gnomon 7
[1931] 351f). On ancient ways of citing sources, see E. Stemplinger, Das Plagiat in der
griechischen Literatur (Leipzig 1912) 177ff. For example, Apollonius of Perga, in Conica
books 1 and 4, mentions his predecessors in a prefatory letter, but this is hardly likely to
have been the case with Speusippus. On the form of expression, cf. Nicomachus (Por.
VP 20 = Iam. VP 30) on the ἀκρόασις of Pythagoras, and ἐσπουδάσθη used of the number
theory by Moderatus (Por. VP 48). ᾿Εξαιρέτως is only attested late (Plutarch, Arrian).
Theo Smyrnacus says (106.10f) that Philolaus had a good deal to say about the number
10 ἐν τῷ περὶ φύσεως, and this could have led to the source attribution.
246
ν. Spurlous and Genuine in the Philolaus Fragments
Further, Nicomachust! ascribes to Philolaus a number system which
represents a development of being in numerical es Ι, point; 2,
line; 3, plane surface; 4, solid; 5, ποιότης καὶ Xpdiors j 6, animation;
7, νοῦς καὶ ὑγίεια καὶ τὸ ὑπ᾽ αὐτοῦ λεγόμενον φῶς; 8, ἔρως, φιλία, μῆτιβ,
ἐπίνοια. This scheme contradicts the report of Aristotle, according to
which the Pythagoreans called the plane surface χροιά, that is, they
were not able to distinguish, even in terminology, between surface
and “color.” In general, this gradation of being, and in particular
the order of geometrical forms, 15 Platonic and not Pythagorean.
In fact, Plutarch cites the very system here attributed to Philolaus, as
Platonic.#2 Nicomachus and Proclus#4 made abundant use of =
“Pythagorean” scheme, which may also have been included in a tepos
λόγος attributed to Pythagoras. | | or
Claudianus Mamertus report of a doctrine on the immateriality ©
the soul comes from a neo-Pythagorean source, where it was right next
to a spurious Archytas fragment;* the contradiction of acon
fragment 14 (in one σῶμα-σῆμα, in the other ae of the soul for
the body) proves that they cannot both be genuine.*
al Th. ar. 74.10 = DK ae ς
43 : 410: cf. above, ch. 1 3, n. 90.
43 aa 3.1.1002a: from μονάς and ἄπειρος δυάς come, τ ae ὙΠ
Mas ἐμ νὰ OH pales oe τ νι ρὲ pe : aoe of Scecuippi
) ya = πονὰς ἀριθμόν, ἂν μὴ τῆς ἀπεῖρου OVO ς
fee ee RE different is the account in De E (390 a pane :
equated with animation. Philo must have known a similar system, since he ¢q 4
Ϊ ἔσθησι u. in Exod. 2.97). |
ue ae Th. ar. 44.1ff, §2.5ff (ch. II 3, n. 110). At 63.256), ς = er
χροιὰ καὶ φῶς (so that this idea is differently placed here) μετὰ τὰ dane τὴ εν Ρ ἢ
διαστάντα, 6 -- ψύχωσις, ἕξις ζωτική, 7 = τελείωσις, διανόησις. : τος ἢ " ν
270.5ff: Πυθαγορείων λόγος, ἀνὰ λόγον ταττόντων σημείῳ μ y pe τ bh ἊΝ
δὲ δυάδα, τῷ δὲ ἐπιπέδῳ τριάδα, τῷ δὲ σώματι τ δρᾷ ἐνὸν τΠ " ἐν: δὲ 4 ae
~ 83 διμνωμένῳ THY ἐξάδα, τῷ δὲ νοερῷ τὴν ἑπτάδα. This is repeated 4 ath
Ke peacaaed pelts νόον φῶς (a trace of Doric dialect). ‘I his : oe es : ᾿ ᾿ a.
interpreted in the context of Neoplatonism (contra Mandolte i a 3 i sae
find pre-Socratic parallels). Delatte (Litt. 201) attributes the Proc emer ἜΠΗ
“Ιερὸς λόγος of “ Pythagoras.” Doubts are expressed by ae ; τ ᾿ : " εἰ Mes
(314ff) justly emphasized the Platonic character of this sytem, : vies ie ae
pretations are arbitrary. This comes out even more clearly in his ana oT ‘ ae Es οὐ
of Speusippus’ system (239ff; criticized by Mondolfo in ae ᾿ ᾿ : ae δ,
important point, relating to the treatment of the soul, by Mer at a P nee a "
46 Philolaus fr. 22, ps.-Archytas p. 47-9 Thesleff, ap. Claud. Mam. : ς ᾿ | ei
ΒΕ. Bémer, Der lateinische Neuplatonismus und Neupythagoreismus Hid . Ἵ ae ὙΠ
μι Sprache und Philosophie (Leipzig, 1936) 143ff. H-Gonipere alice uae
Hermes 1932, 156; but the ae cclasenanecarea ne ia sre
. ahiections of Plato to the soul-harmony ‘doctrine &5c. "
ὌΠ ἀπ eee is stressed by Mondolfo in 7M 377. Of course both fragments
could be spurious.
Hl, PHILOLAUS
Individual apophthegms, in the tradition, are always problematical.
The sentence cited by Clement about the punishment and burial of
the soul in the body is based on παλαιοὶ θεολόγοι τε καὶ μάντεις, but this,
and the close correspondence with Aristotle's Protrepticus, are in
themselves cause for suspicion.4” It was only by a misunderstanding
that Bocckh thought this saying of Philolaus was cited in Plato's
Gorgias.*8 It seemed so natural to reconstruct Philolaus’ doctrine from
the passage in the Phaedo that names him, that nothing more than this
seems to lie behind the explicit statement of Athenagoras about the
dpovpa.4® The saying of “‘Philolaus” in praise of number (fr. 23) is
47 The word θεολογία is first attested at Pl. Rep. 3794; θεολόγος is common in Aristotle.
Cf. Jaeger, Theol. 4; Vlastos, PhilosQ_1952, 102 ἢ. 22. The latter points out that parallel
formations like perewpoAdyos, φυσιολόγος, μυθολόγος are common, and that ἀμφὶ θεῶν
λόγος occurs in Emp. fr. 131. V. Goldschmidt, “Theologia,” REG 63 (1950) 20-42,
ignores the Philolaus fragment. With fr. 14, μαρτυρέονται δέ μοι of παλαιοὶ θεολόγοι
τε καὶ μάντιες, compare Philo Op. 100: μαρτυρεῖ δέ μου τῷ λόγῳ Φιλόλαος, and esp. Arist.
fr. 60: καθάπερ φασὶν οἱ τὰς τελετὰς λέγοντες... τοῦτο γὰρ θείως οἱ ἀρχαιότεροι λέγουσι
(lam. Protr. 47.23), or in Latin, “‘ut interdum veteres illi sive vates sive in sacris initiisque
tradendis divinae mentis interpretes . . . aliquid vidisse videantur” (Cic. Hortensius fr. 95
Mueller == Aug. C. Iul. 4.15.78). Likewise, with fr. 14: ὡς διά τινας τιμωρίας, Arist.:
‘ob aliqua scelera”’ (Cic.) or διδόναι τὴν ψυχὴν τιμωρίαν καὶ ζῆν ἡμᾶς ἐπὶ κολάσει μεγάλων
τινῶν ἁμαρτημάτων (Iam. Protr. 47.2 5); fr. 14: ἃ ψυχὰ τῷ σώματι συνέζευκται; Arist.:
ἡ σύζευξις. .. πρὸς τὸ σῶμα τῆς ψυχῆς (lam. Protr. 48.2f). By “priests of the mysteries”
Aristotle clearly means the Orphics. But at Pl. Crat. 400c the expression σῶμα-σῆμα is
distinguished from their view (Wilamowitz, GldH II 199; Thomas 51; Dodds, Irr.
169 n. 87); and we may suppose that, if it is not Orphic, it is likely to be Pythagorean.
The ostensible Philolaus, however, contaminates the two and leaves us to guess whether
he regards Pythagoras as an “‘ancient seer.” The σῶμα-σῆμα idea was familiar in later
times (Cic. Rep. 6.14, Macrob. Somn. Sc. 1.10.9f, Serv. Aen. 6.127, 439).—Clearchus,
fr. 38, quotes a similar expression from the Pythagorean Euxitheus; but we do not know
whether this was a historical character (Wellmann, RE VI 1539; Wehrli 59).
48 At Pl. Gorg. 493a-b, Socrates—Plato cites a σοφός ( = ὁ πρὸς ἐμὲ λέγων, 493b), who
teaches that life is death and the body a grave, and bases this doctrine on allegorical
interpretation of what a μυθολογῶν κομψὸς ἀνὴρ, ἴσως Σικελός tis ἢ ᾿Ιταλικός had said.
The ‘‘mythologer” and the exegete must be distinguished from each other, though this
is often forgotten. (The wrong interpretation is in Boeckh 183ff, Bywater 47ff, Schaar-
schmidt 6f, Carcopino, Bas. 285ff, Thomas 52f, Long 74f. Even Wilamowitz, Platon
II 8y, does not draw the line sharply enough between the “mythologer”’ and the inter-
pretation. For the correct solution, Frank 298ff, Wuilleumier 571, Linforth, UCPCP
1944, Dodds, Irr. 225 n. 5, Gorg. 296ff.) The “‘mythologer” spoke of Hades and the
punishment of the uninitiated, of carrying water in a sieve—a teaching of the mysteries
widely known in the sth century (Polygnotus’ painting in the lesche of the Cnidians at
Delphi: Paus. 10.31.9, 11; south Italian vases, Méautis 78f; A. Rumpf, Misc. Ac. Ber.
IE 2 [1950] 41f; cf. Wuilleumier 298f, §52; an allusion at Pl. Rep. 363c). The myth may
have formed part of an Orphic katabasis; that the author was “perhaps a Sicilian or
Italiote”’ is phrased like a conjecture, but alludes to the Pythagoreans (cf. above, ch. II 3;
Σικελὸς κομψὸς ἀνήρ is an allusion to Timocreon fr. 4 Ὁ [Wilamowitz, Platon II 89).
That the one who offered the interpretation was himself a Sicilian or Italiote (as Frank
secks to show, go, 29, n. 210) does not follow; so that it is hard to support the conjecture
that Philolaus is intended. Carcopino, Bas. 287f, finds the same “tics d’expression”” here
as in fr. 17 and Arq ‘Th. ar. 81.15; but the basis for such an assertion is too small.
248
2 Spurious and Genuine in the Philolans Fragments
as spurious as its neighbor attributed to Hippasus.°° Through a mis-
understanding of Philo, Philolaus is credited with a scemingly mono-
theistic saying which Lydus ascribes to Onetor.=! Philolaus equated
the number 7 with Athena, the ‘“‘motherless,’ and this agrees with
Aristotle.52 The thought that the number 2 is “the consort of Cronus’
(Rhea) may belong to the same tradition.» There is no reason to
suspect the authenticity of the apophthegm passed along by Plutarch,”
according to which geometry 1s the ἀρχὴ καὶ μητρόπολις τῶν ἄλλωι
μαθημάτων, or the thought that the κριτήριον is the λόγος ἀπὸ τῶν
μαθημάτων περιγινόμενος, though the terminology 1s influenced by the
more abstract thought of later times, reflected in the doxographers.®
There is no occasion to attribute to Philolaus the definition of ἁρμονία as
ΝΥ , , 56
πολυμιγέων ἕνωσις καὶ δίχα φρονεόντων συμφρόνησις.
I
We cannot regard the word σοφός (cf. above, ch. I 3, n. 157) or the use of allegorical
interpretation as a sure indication of Pythagoreanism. There are, however, some indica-
tions of Pythagorean influence: “ Hades”’ is located in man’s lifetime, as in Empedocles
(cf. ch. I 3, n. 80; ΙΝ 4); μνήμη and πίστις are thought to hold together the forces of the
ψυχή (cf. ch. II 4). The background to be assumed is a writing similar to the one intro-
duced to us by the papyrus from Derveni. |
49 Fr, τς = Athenag. 6; Pl. Phd. 61d ff; cf. above, ch. Ill 1, n. 55; Frank 295ff. .
60 Fr, 23, and Hippasus DK 18.11; the two citations are neighbors at Iam. In Nic.
10.20ff, and, later, Syrian. Met. 123.6ff, 142.21ff. There was no writing of Hippasus
(D.L. 8.84). The style is imitative of the Timaeus. ; .
51 Fr. 20 = Philo Op. 100, Lydus Mens. 2.12. P. Boyance has shown that Lydus is not
dependent on Philo (REG 76 [1963] 91). The form of the name Onetor 1s suggested by
the MS reading ὁ νήτωρ recorded by Thesleff, Texts 140.21 n., though he himself prefers
ee i ionale: ov a ὧν ἐν τῇ
52 Alex. Met. 39.3ff = Arist. fr. 203, with the same rationale: οὔτε te τινα τ ἢ
“ 3 ~
δεκάδι ἀριθμῶν 6 ἑπτὰ οὔτε γεννᾶται ὑπό τινος αὐτῶν... ἀμήτωρ... παρῦὕενος.
68 Fr. 208. Χοποοαῖθβ, too (fr. 1§ = Aét. 1.7.30) associates duds and μήτηρ θεῶν.
Doubtless Rhea is here associated with ῥέω, and this is how she comes to provide a name
for the dévaos ὕλη (Xenocrates fr. 28), which becomes ἀόριστος δυάς. But Rhea also meets
mata (above, ch. II 4, n. 32). ae.
us ΠΣ arte a conv. §.2.1.718e, where the MSS have φίλαον. It is ate
that, correctly in consideration of the history of mathematics, geometry is placed first;
in the Platonic hierarchy arithmetic stands first (cf. above, ch. Π I, n. 14).
55 The passage printed in DK 44A29 is misleading. Posidonius (ap. Sext. Emp. ae
7.92ff) is discussing opinions on the κριτήριον. Anaxagoras named een edi ; e
Pythagoreans, more explicitly, the λόγος that comes from μαθήματα “as P Αὐτὴν =
said.” The thought is added (connective τε) that the Pythagoreans spoke of a re ee a
of this λόγος with the entirety of nature; and in the next sections (92-109, not in δ
this idea is developed at length. But the reference for this second idea is Lae: r.
109), not Philolaus; so that only the first part, about learning through μαθήματα, belongs
to him. The basis is A7a and fr. 4.
60 Er, 10, in Nicom. Ar. 2.19, Theo Sm. 12.10. As with number, there must have been
pithy sayings of the Pythagoreans current about harmony; cf. Cone ap. ἘΝῚ
Quint. τ p. 3 Μ. and above, ἢ. ςο. Ἕνωσις sounds post-Platonic (cf. the ue inition of μίξις
in Arist. Gen. corr. 328b22.) Boeckh (61) first attributed the fraginent to Philolaus.
249
Ht, PHILOLAUS
The decisive group of fragments, in which Philolaus speaks about
being which is “limiting” and “unlimited,” as well as about number
and harmony, has received very little attention; and, in particular,
the opponents of their authenticity have not attempted any thorough
analysis.5?7 Yet only such an analysis can determine whether what we
have is a post-Aristotelian “forgery” or a pre-Socratic composition.
In the former case, it should be relatively easy to explain it in the
categories of Platonic, Aristotelian, and post-Aristotelian thought; in
the latter, we should expect that the familiar concepts and formulae of
later times would prove inadequate, and that an approach would be
possible only through the special characteristics of other pre-Socratic
thinkers.
¢ 4 32. 3 - / ξ ,ὔ 3 3 ,
A φύσις δ᾽ ἐν τῷ κόσμῳδϑ ἁρμόχθη ἐξ ἀπείρων τε καὶ περαινόντων,
" Ὁ, € / ~
καὶ ὅλος <6> κόσμος καὶ τὰ ἐν αὐτῷ πάντα (B 1).
"A ? , 37 > 7 Ἄ é “uO Ἃ , ,
νάγκα τὰ ἐόντα εἶμεν πάντα ἢ περαΐνοντα ἢ ἄπειρα ἢ περαίνοντά τε
, av . a A 7 Ἅ ,
καὶ ἄπειρα ἄπειρα δὲ μόνον «ἢ περαίνοντα μόνον» ov Ka εἴη. ἐπεὶ
“ f »ν F >
τοίνυν φαίνεται οὔτ᾽ ἐκ περαινόντων πάντων ἐόντα οὔτ᾽ ἐξ ἀπείρων
; Ἂς - ea > ,
πάντων, δῆλον τἄρα ὅτι ἐκ περαινόντων TE καὶ ἀπείρων 6 TE κόσμος
᾿ ‘\ ? 3 ~ ἦ \ 4 / ς
καὶ τὰ ἐν αὐτῷ συναρμόχθη.... (Β 2). Περὶ δὲ φύσιος καὶ ἁρμονίας
τ ὃ μὲ 59 ς 4 > A - i 323} μὲ \ ? Ἁ A ς
ὧδε ἔχειδϑ, a μὲν ἐστὼ τῶν πραγμάτων ἀΐδιος ἔσσα καὶ αὐτὰ μὲν a
7 , A ~
φύσις θείαν re® καὶ οὐκ ἀνθρωπίνην ἐνδέχεται γνῶσιν πλήνδ᾽ ya ἢ
¢ ᾽ / 3 ᾿ \ ~ > ~
ὅτι οὐχ οἷόν τ᾽ ἦν οὐδενὶ τῶν ἐόντων Kal γιγνωσκομένων ὑφ᾽ ἁμῶν
51 The most extensive is that of Rothenbiicher (668); but his conclusion, that the whole
complex is “absurd and not Pythagorean,” reveals his failure to understand it.—By water
only cites some verbal reminiscences of pseudo-Pythagorica. Frank (302ff) paraphrases.
Scoon (GrPh 133ff) treats the matter more fully; see also A. Burns, C&M 25 (1964)
Y3-128.
4 Heidel’s conjecture a φύσις δὲ τῶ κόσμω (AJP 1907, 79) is mistaken; cf. Anaxagoras
fr. 8: τὰ ἐν τῷ ἑνὶ κόσμῳ; Diogenes of Apollonia fr. 2: ra ἐν τῷδε τῷ κόσμῳ ἐόντα
(similarly Hippoc. Nat. hom. 7, VI so L.); Hebd. 1: “mundi forma sic omnis ornata erat
corumque quae insunt singulorum” (τοῦ κόσμου ἡ ἰδέη ὅλον καὶ τῶν ἐν αὐτῷ ἐόντων
ἑκάστων, Pfeiffer, Sterngl. 31.2); ἃ φύσις τῶ κόσμω would in itself be suspicious (the other
way around in Eur. fr. 910: ἀθανάτου φύσεως xéopov...); cf. also Kahn 228f. On
φύσις see 1). Holwerda, Commentatio de vocis quae est φύσις vi atque usu praesertim in
Graccitate Aristotele anteriore (Diss. Groningen, 1955). The use of φύσις to mean the totality
of ἐόντα is common in the time of Philolaus (as Euripides in the passage cited above).—-On
ἁρμόχθη cf. Emp. fr. 107: ἐκ τούτων γὰρ πάντα πεπήγασιν ἁρμοσθέντα, and the Derveni
papyrus, col. 17.
δ Cf. Hippoc. Morb. init.: mepi τῆς ἱρῆς νούσου καλεομένης ὧδ᾽ ἔχει.
80 Sic MS F; ya Diels, θεία ἐντί Badham, Wachsmuth. For re... καὶ οὐ in antitheses cf.
Hide. 8.81, Soph. EL 885, J. 10. Denniston, “Uhe Greek Particles (Oxford, 1954?) $13.
“MSS πλέον, retained by Diels and Scoon (354), with the thought that it modifies
γνῶσιν (“no more... than"); but one would expect ἐπὶ πλέον or πλέονα. The conjecture
of Badham, followed by Wachsmath and Rostagni (Verbe $0 n, 2), is supported by the
fact Chat πλὴν ἤ isa common combination (cp. dt. διτα ας Ar. Neb, 364).
+, Spurious and Genuine in the Philolans Fragments
γεγενῆσθαι μὴ ὑπαρχούσας τᾶς ἐστοῦς τῶν πραγ βατῶν: ἐξ ὧν
συνέσταβϑϑ ὁ κόσμος, καὶ τῶν περαινόντων καὶ τῶν ἀπείρων. ἐπεὶ δὲ ται
ἀρχαὶ ὑπᾶρχον οὐχ ὁμοῖαι οὐδ᾽ ὁμόφυλοι ἔσσαι, Ἴ ἢ ἀδύνατον ἧς κα
αὐταῖς κοσμηθῆναι, εἰ μὴ ἁρμονία ἐπεγένετο, Οὐ ον hog τροπῷ:
ἐγένετο. τὰ μὲν ὁμοῖα καὶ ὁμόφυλα ἁρμονίας οὐδὲν εἐπεθεουτο, τὰ δε
ἀνόμοια μηδὲ ὁμόφυλα μηδὲ ἰσοταχῆ,“ ἀνάγκα τὰ τοιαῦτα ἁρμονίᾳ
συγκεκλεῖσθαι, at μέλλοντι ἐν κόσμῳ κατέχεσθαι... (Bb )
The φύσις in the cosmos has been put together harmoniously from
unlimited and limiting (constituents), both the whole cosmos and
all the things in it (fr. 1). . -- "
Existing things must be, all of them, either limiting, or unlimited,
or both limiting and unlimited; but they would not be unlimited
only <or limiting only(?) >. Since, however, they are clearly neither
made of limiting (constituents) only nor of unlimited only, it 15
therefore obvious that from both limiting and unlimited (constitu-
ents) the cosmos and the things in it were harmoniously put to-
gether . . . (fr. 2). This is the situation about φύσις and harmony: δ
being (ἐστώ) of things, which is eternal, and φύσις itself admit o
62 F the only authoritative codex, has γεγνέσθαι. Mullach altered οὐθενί to οὐθέν,
Usener and Diels to οὐδὲν τῶν ἐόντων καὶ γιγνωσκόμενον .. - YO γενέσθαι. The ae
to this is that the passive is never expressed elsewhere, 1 believe, by hist τ an
present passive participle. (NT Rev. 16.10 has a comparable one ἢ =
the perfect ppl.: ἐγένετο . - - ἐσκοτωμένη. Elsewhere γίγνεσθαι is used with t aes ive
ppl.; cf. Blass-Debrunner-Funk, A Greek Grammar of the New ee ἐν πρὸς
1961] ὃ 359.) Therefore the MS reading is to be retained (with Rostagni, ley 9 50n. se
Scoon, GrPh 354ff). Being (ἐστώ) is a condition not of knowledge, but of the sae
world of ours. On the conjecture γεγενῆσθαι, cf. the pert. pass. inf. a little furt er δὴ
and Antiphon fr. 58, τό γεγενῆσθαι, for an unalterable situation; aside from that, it avoi :
the undesirable ye. One might also think of a Doric γενέσθην. On the ale Οὐδι σεν ς :
Hippoc. Vet. med. 15: τὰ γιγνωσκόμενα are equivalent to aes facts of the world o
i . Democritus fr. 165: ἄνθρωπός ἐστιν ὃ πάντες ἴδμεν. . .. ;
ie ater cf. Eur. fr. 910, Diogenes of Apollonia fr. 2, the Derveni ig? bi Ee
17.3, Arist. Met. 99026, ὑπάρχειν and ὁμόφυλα Hippoc. Nat. hom. 3, ΝῚ 38 L.; οἵ. Vem
i 135, DK If 115.2. .
we Ἔ MS reading ou be understood by comparison with Leucippus oe ee
n. 101), but all the editors emend. Heidel (AJP 1907, 78) proposed tel w ig bik
accepted by DK, but this word is late attested elsewhere. From actual pre-soc
vocabulary a possibility is ἰσοπαλῆ (Meineke). Cf. below, n. 74. ate
65 The MS reading is τὰ τοιαῦτα ἁρμονίαις συγκεκλεῖσθαι, ἢ εἰ... The ees reading
τᾷ τοιαύτᾳ ἁρμονίᾳ, and Meineke further altered ἢ εἰ to ai and Dee meee - an wea
but εἰ μέλλει is so common a combination that it ought not to be altere . ᾿ ve
fr. 264, Pl. Rep. 376b, 416d, and passim; similarly Eur. Or. 292, fr. 924). Ont
hand τᾷ rotatra .. . οἷᾳ is an unusual and harsh locution; ὁ τοιοῦτος refers Ἢ a ᾿ aa
terization previously mentioned or known (see Kiihner—Gerth II 1. 10: ), δὶ er ᾿ an ee
following οἷος. Anyway, “such a harmony” would be an odd phrase; the author oe
talking about any particular organization. Thus it scems best to keep τὰ beats " ἦν
can refer to ἀνόμοια ctc., and to punctuate before ἀνάγκη, even though t " crea "
asyndeton. (A sentence is frequently introduced by avayxy, as oe i a ω
Antiphon fr. 44 |DK 1] 354, col. 1 line 14], Zeno fr. 1, Melissus i 7, Philolaus fr. 2.
For συγκεκλεῖσθαι sce Emp. fr, 27: ἁρμονίης πυκινῷ κρύφῳ ἐστήρικται.
24!
II. PHILOLAUS
divine
ἀρ δεν a in sae knowledge, except that it was impossible
of the things that exist and are r
dee eres : ecognized by us to come to
not the being (ἐστώ) of
the things of which th
cosmos was composed, both the limiti ee
: : imiting and the unlimited. A
since [ΠΗ] ἰρχαί Ι ene
ae ae beginnings (ἀρχαί) were not alike or of the same kind, it
ave been impossible for them to b | |
| e put together ha i-
ously if harmony had no : on
t supervened—however it hat i
to be. It is not things th are eee ες
gs that are alike and of th ki
ἢ 3 e same kind that need
armony, but things unlike and different and of unequal speed;
such things must be bonded togeth
held together in a cosmos... (f6). by harmony, if they are to be
». ᾿ i caer: here is the really heavy-handed insistence with
ch the leading ideas—“limiti imi
elena g ideas—‘‘limiting and unlimited,” “cosmos,”
Bs ny —are repeated. But far from giving cause for suspicion,®
: rg procedure makes the impression of a genuine effort on
ought, anxious to keep hold of the i
| e important points, and not seck
τὶ | seck-
: A ο εἰ τὶ a display with carefully learned or borrowed fustian. This
s the typica style of the pre-Socratics; “τῆς amount of tedious repeti
ti » ν᾿ - * ᾿
= a in Anaxagoras, is incredible.’’®? The very opposite of this is
the high-flown style, derived fi
: ed from the Timaeus, of a pi iti
he a , of a piece of writing
an ari world soul (A2 1). Here one notices similarities of
guage to Anaxagoras, Euripides, Empedocles, Diogenes of Apolloni
and the Hippocratic writi acces ἀν τὶ
a) Ρ writings. Then there is the surprising δέ in the sen-
Ἢ t : : quoted by Demetrius of Magnesia as the first in the book
έ . . ν ᾿
in the first sentence, connecting it with the title, is found in several
᾿ς works of the fifth century ;98 and this feature also guarantees th
t Ά i . « . Ξ
᾿ e ἐν ΜΕΝ Wilamowitz said of the style of the fragments, “Ihre
prache ist entschieden altertiimlich, macht den besten picruck ae
66 Frank 305 - “Thi Ξ ἜΝ
: is sterile repetition of the same i
. me ideas, while th
rogre : ἢ e argument
progress at all, this continual manipulation of the same phrases, is i ee
a second-rate mind.” , is intolerable; it betrays
87 Diels, SBBin 18
ΟἹ, 580 n. 2, on Anaxa fr. 6: .
ion (referring to Parm. fr. 8.33f); also Melissus fr. 2, Diog. A 1 Departs
fr. το]. . 2, Diog. Apol. frr. 2, 5, Democr.
68 Her. fr. 1, I : :
Septim. 1, Ὁ ie oe Pol. ᾿ Hippoc. Mul. 1, Nat. nul. 1, Foet. exsect. 1
ee ΕἸ, . Ap. 1, Ove. 1.1, Critias fr. 46. First came eee
(e.g., Φιλόλ oe ad 46. First came the author and
follawed ae τ, idee ὧδε λέγει"). Ina continuous text ὧδε is
: t alse COLLIE vy € is Hi poe A ’ 7% ἣ ; : ᾧ
Staate der Athener (Beri as Hippoc. Aer. 13. Cf K. Gelzer, Die Schri
fr Saiki cea dee 1937) roof, Boeckh mistakenly concluded from the 4 stat
ot be the beginning of the book (4.41). - This usage, it is true, wa si
᾿ ne ee ee *, was often
imitated in later times, eg. by P
ι cg. by Pollux (I. Bethe, Hermes
® Wilamowitz, Platon IL go. Eee
2. Spurious and Genuine in the Philolans Fragments
The author’s basic idea is the division of ἐόντα or πράγματαϊ" into
ἄπειρα and περαίνοντα; the breach which this opens up in the cosmos 1s
healed by a ἁρμονία. There are obvious similarities with what Aristotle
reports, but also characteristic differences. In the first place we have
περαίνοντα (“limiting [things], or “shat which limits”) instead of
πέρας (‘the limit”’) or τὸ πεπερασμένον (“the limited’’). Though it has
model in the not infrequent expression περαίνειν πρός τι (“ border on
something’), still this absolute use 15 notable. But it 15 comprehensible,
or even necessary; a “limited” presupposes something else which is
“limiting” oF “limit-setting.” The necessary complement to what is
unlimited is the existence of “things” that “make limits,” which bear
the principle of limitation as an active power.”? Now in one of Aris-
totle’s dialogues, in a clearly “Pythagorizing ” passage dealing with the
perfection of musical harmony, he uses the phrase ἡ περαίνουσα φύσις
for the nature of an odd number.”? Hesychius s.v. πέραινον (i.e. περαῖνον)
says οἱ Πυθαγορικοὶ τοὺς περιττοὺς ἀριθμούς, and formulations like this
are found in Nicomachus.”* The subject of this passage of Philolaus is
not primarily numbers, but “things,” and the antithesis περαίνοντα καὶ
ἄπειρα is used in ἃ broader and more nearly original sense than in the
definition given by Hesychius.
Significantly, the “principles,” in spite of the occurrence of the
suspicious word ἀρχαί,13 are never designated by an abstract expression
like τὸ πέρας, τὸ πεπερασμένον, τὸ ἄπειρον, OL, aS in the formulation of
Aristotle cited above, ἡ περαίνουσα ¢vous—or, as in the Philebus,”
τὸ πρᾶγμα in the general sense is equivalent to ov elsewhere: Democr. fr. 164 (cf. Hdt-
2.42), Gorg. Hel. 17, 18, Pal. 24, Epicharm. fr. 3, Euenus fr. 8 Diehl (cf. Theog. 472)"
Obviously χρῆμα is the older expression, and is always used by Anaxagoras and Melissus
(Aét. 1.3.5 gives the gloss Anaxagoras fr. 1, χρήματα λέγων τὰ πράγματα); cf. Diels,
SBBin 1884, 350 ἢ. 1.
71 This is emphasized by Reinhardt, Parm. 65 n. I, and Scoon, GrPh 14off; Melissus
fr. ς: περαίνειν πρός τι, Hippoc. Vict. 1.10, Pl. Meno 76a: σχῆμα εἰς... ὃ τὸ στερεὸν
mepaiver.—Aristotle deals, in his own system, with ἄπειρον and πεπερασμένον (Phys.
202b30ff), as had Zeno, fr. 3.
12 Fr, 47; cf. above, ch. 1 3, n. 161. The conjecture περισσῆς, still printed by Ross,
is to be rejected because of the Hesychius passage.
73 Odd numbers are περαινούσης οὐσίας, Th. ar. 62.15; square numbers made with
odd γνώμονες (above, ch. I 2, 1. 27) are ὡρισμένοι TE καὶ περαίνοντες, Nicom. Ar. 2.18.3
(where the citation of Philolaus fr. 2 follows in the next section).
74 A similar plural Hippoc. Morb. VIL $42, 582, 590 L. (Keus 128); cf. also Pi, Neat.
18. It is significant that in Philolaus a past tense is used (differently from ps.-Archytas
69.6 Thesleff; ἀνάγκα δύο ἀρχὰς εἶμεν τῶν ὄντων). It would be convenient to emend
ἐπεὶ δέ τε (sic F) ἀρχὰν ὑπᾶρχον οὐχ ὁμοῖα οὐδ ὁμόφυλα ἐόντα, ἤδη ἀδύνατον ἧς κα αὐτοῖς
(sic Ε) κοσμηθῆναι ....ὄ (on ἀρχάν see Anaxagoras frr. 6, ta). But ἐόντα could not be
changed to ἔσσαι by an ordinary slip.
τὸ P]. Phib. 240, c6 TBs. Por ancient connection of Philolaus and the | Hhilebus, see above,
ch. 14, ne 14.
Il. PHILOLAUS
ἡ τοῦ ἀπείρου φύσις, though this is the regular practice in the later
tradition.’6 “Limit” and “‘unlimitedness’’ are not isolated as entities in
themselves, congealed into an abstract substantive or hypostasized as
intangible substance, but they are thought of as scattered or deployed,
so to speak, in individual things, wepatvovra or ἄπειρα. This is a basic
difference between the thought of the pre-Socratics and that of the
Platonic and Aristotelian schools, a difference brought out especially
by Cherniss.”7 By a word like θερμόν or ψυχρόν, for example, a pre-
Socratic thinker does not mean an abstract quality, or an οὐσία, but
quite concretely the sum of particular things characterized by the word.
Only in Plato’s dialectic was the foundation laid for separating
qualities and quantities from objects and regarding them αὐτὰ καθ᾽
αὑτά. For later ages these distinctions came to seem self-evident. By
saying not ἀνάγκα δύο ἀρχὰς εἶμεν τῶν ἐόντων, τό TE πέρας καὶ TO
dretpov,’® but ἀνάγκα τὰ ἐόντα εἶμεν πάντα ἢ περαίνοντα ἢ ἄπειρα ἢ
περαίνοντά τε καὶ ἄπειρα, Philolaus shows his affinity, in expression
and thought, with the pre-Socratics and his difference from all that
is Platonic and post-Platonic. From this point of view a forgery is
unlikely, since what is involved is not the kind of technical terms or
verbal flourishes that are easy to imitate, but the way in which the
mind goes about trying to understand reality.
The division of “things” into “limiting” and “unlimited” is of
course more abstract than an analysis in terms like “warm,” “cold,”
“dry, and “moist,” where reference is made to areas of immediate
sense perception. There is no indication what kind of specific experience
is implied in the idea of “limit” and “unlimited.” The author does
give what looks like a hint, in the continuation of fragment 2: δηλοῖ
δὲ καὶ τὰ ἐν τοῖς ἔργοις. τὰ μὲν yap αὐτῶν ἐκ περαινόντων περαίνοντι, τὰ
δ᾽ ἐκ περαινόντων τε καὶ ἀπείρων περαίνοντί τε καὶ οὐ περαίνοντι, τὰ δ᾽
ἐξ ἀπείρων ἄπειρα φανέονται. But this sentence is couched in quite
general terms, and it is impossible to tell what specific sense the word
ἔργα has;’® all we learn is that the “limiting” and the “nonlimiting”
6 Aét. 1.3.10 (DK 44A9), Damasc. Princ. I 101.3 Ruelle, Arist. passim.
7 Pres. 3758, JHI 342ff.
8 Cf. ps.-Archytas 19.5 Thesleff (above, n. 74).
19 Boeckh (50) translated “ Bauwerke,” Diels (Vorsokratiker!) “Acker,” Scoon (GrPh
138) advanced the idea that ἄπειρα are mountain meadows without definite boundaries.
Newbold (186ff) essayed a geometrical answer, Becker (QSt 4, 154f) an arithmetical
(ἔργον is the product; “odd” x “odd” “odd,” ete.). Heidel (AJP 1907, 773 Proc.
1913, 730) has “nothing but the bald assertion that observation or knowledge confirms...”
Similarly, W. Nestle (Philologus 67 [1908] s44f). CA Anaxagoras B7: μὴ εἰδέναι... μήτε
λόγῳ μήτε ἔργῳ. Prank (307 8, 1) points out that expressions like δηλοῖ τὸ ὄργον occur in
2. Spurious and Genuine in the Philolaus Fragments
that “things” do is dependent on the “limit-forming” and “un-
limited”’ constituents that go into their makeup.
It is very easy to understand “limit” as a formative principle,
and “‘unlimitedness” as a material principle. But in considering a
question of authenticity we must differentiate between philosophical
interpretation, which seeks to understand the author better than he
understood himself, and philological interpretation, whose first duty is
to understand and place historically what the author put down. The
thought that god gives form to formless matter, by imposing “limits,
περατοῦν τὴν ὕλην ἄπειρον οὖσαν, is characteristic of the Platonists;®°
that it does not go back to the Pythagoreans, and that it was not
an accomplishment of theirs, as is often thought, to conceive for the
first time, in explicit terms, of a formal principle, follows from the
exposition of Aristotle, according to which the form-matter dichotomy
is not applicable to their number theory. If the situation were different
in the Philolaus fragments, this would be a serious cause for suspicion;
but any such interpretation is impossible. Boeckh equated the “limit
with the One, the “unlimited” with the Indefinite Dyad, so that the
difference between Philolaus and the Platonic system disappeared.”
On this basis, Frank was able to maintain® that Philolaus agreed with
Plato, against the testimony of Aristotle. But the text of Philolaus 1s
against this. The One is itself a ἁρμοχθέν, and is therefore not simply
περαῖνον.88 After all, “limiting (things)”’ (wepatvovra) and “unlimited
(things) ” (ἄπειρα) stand side by side, both in the plural, and there is no
indication that in the process of “harmonization” (ἁρμόζειν) the
“unlimited” disappears; nor is there any suggestion of the one
affecting the other, in the way that “form” must affect “matter.
The ideas of ἐστώ (“being”) and ἁρμονία complicate the matter
I
Aristotle (Mete. 349b35, cf. Met. 1086bs5); but they are also in use earlier, as Hippoc.
Flat. 5: πρὸς αὐτὰ τὰ ἔργα... πορευθεὶς ἐπιδείξω... . On the idea, cf. Hippoc. Nat. puer.
VII 498 L.: ὅσ᾽ ἀπὸ πυκνῶν ἐγένετο πυκνά ἐστι καὶ ὅσα ἀπὸ ὑγρῶν Dypd.
80 Plut. Quaest. conv. 8.2.3.719c. Cf. the construction of elements in the Timaeus and
in general the system of the One and the Indefinite Dyad. "
81 Boeckh 54f, followed by DK I 407 π.: “Grenze (Form) und Unbegrenztheit (Stoff)”;
I 409, in the translation, “diese Prinzipien (1 und 2).”” Contra, above, ch. I 2.
82 268, 308 ἢ. 1, 309 n. 1; accepted by Cherniss, Pres. 391. The fact that the octave,
which is also called dppovia, shows the ratio 2:1, has nothing to do with the Indefinite
Dyad. ᾿ ou, ; :
88 Fr, 7: τὸ πρᾶτον dppoabdr, τὸ ἕν, ἐν τῷ μέσῳ τᾶς σφαίρας ἑστία καλεῖται. Phough
τὸ & looks suspiciously like dittography, the combination of the πρᾶτον ἁρμοσθέν with
the πρῶτον ἕν συστάν in Aristotle (Met. ro8ob20, cf. rogtar2ff) still stands quite apart
from the problem of authenticity.
44 Ay in Plutarch (above, n. 80) and in Scoon's interpretation (below, 1. 85).
Hil. PHILOLAUS
still further. We cannot simplify it by equating ἐστώ with ἄπειρον and
ἁρμονία with περαῖνον. The ἐστὼ τῶν πραγμάτων... καὶ τῶν περαινόντων
καὶ τῶν ἀπείρων is prior to the differentiation of ‘‘limiting” and “un-
limited,” and belongs to onc of these as much as the other, and “har-
mony, when it “comes,” applics to both of the “beginnings, which
are not alike nor of the same kind.” Nor can ἐστώ be thought of as
ὕλη or ἁρμονία as formal cause.® ᾿Εστώ is not ἄποιος ὕλη, but already
differentiated into “limiting” and “unlimited,” and “harmony”
comes along later; it would not be necessary at all, if it had not been
for the antithesis of “limiting”’ and “unlimited.” There is no hint here
of a division of being into form and matter, which must always be
thought of in relation to each other—nothing that goes beyond the
pre-Socratic picture of the cosmos as the proper ordering of something
previously chaotic.
The word ἐστώ is not to be understood as a material principle in the
Platonic and Aristotelian manner. The only thing that is said of it is
that ὑπάρχει. Being is. This is the fundamental position of the Eleatics,
and it is from this point of view, rather than that of later, abstract
conceptualization that the sense comes clear. Since Parmenides, all
philosophers had spoken of an eternal being,®¢ and the Sophist Antiphon
used the word ἀειεστώ.37 This concept of being had been the subject
of heated controversy since the time of Zeno, Melissus, and Gorgias.®®
85 As by Scoon, CP 1922, 354, GrPh 1308, esp. 145 n. 30, who equates the unknowable
ἐστώ with ἄπειρον and πέρας with number, the principle of knowability, and ἁρμονία.
The ἄπειρα, he thinks, are rudiments, which the ἁρμονία has not yet fully grasped. Howald
was able to cite Scoon in order to bolster his argument that the fragments are infected
with Platonism (p. 67; cf. Burnet, EGP 285 n. 3). A suspicious resemblance between the
unknowable ἐστώ and the Aristotelian ἄποιος ὕλη, ἄγνωστος καθ᾽ αὑτήν, is seen by Schaar-
schmidt 66, Bywater 34f, Frank 308, and Raven in KR 310f. But for Plato and Aristotle
ὕλη is unknowable in principle, and not the object of some higher, divine knowledge.—In
ps.-Archytas p. 19.19 Thesleff, ὠσία-ἐστώ is certainly the Aristotelian material principle,
contrasted with pop¢dc).—Rostagni, Verbo 4off, interprets ἐστώ as “essentia,” ideal Being,
supernatural “‘ Unity.” This, too, is conceived from a Platonic point of view. The contra-
dictions among the interpretations show that Platonic principles are not applicable to
Philolaus 6.
86 Melissus frr. 1, 7, 4, Empedocles fr. 17.3 = 26.12, Diogenes of Apollonia frr. 7, 8
(ἀήρ as ἀίδιον καὶ ἀθάνατον σῶμα, which does not deny the perishability of the cosmos,
Ato), Gorg. fr. 3 §§68f (hypothetical: εἰ ἀίδιον τὸ dv).—Eur. fr. 910; cf. Anaxagoras fr. 17.
8? Antiphon fr. 22 = Harpocr. s.v. (It is explained as τὴν ἀιδιότητα Kal τὸ ἐπὶ τῶν
αὐτῶν dei ἑστάναι.) On the formation of the word, cf. εὐεστώ, Acsch. Sept. 187, Ag.
647, 929, Hdt. 1.85, Democr. fr. 2c; ἀπεστώ, Hdt. 9.85; συνεστώ, Hdt. 6.128. It is ob-
viously an Ionic formation; Pl. Crat. 4o1c shows that the Doric is wala or ἐσσία. Iam.
VP 162 cites ἐστώ as one of Pythagoras’ neologisms, along with κόσμος, φιλοσοφία,
τετρακτύς. Ps.-Archytas, pp. 19.§-20.17 Thesleff, used ἐστώ alternatively with ὠσία.---
On αὐτὰ μὲν a φύσις in Philolaus tr. 6, cf. κατ᾽ αὐτό in Parmenides fr. 8.58; similarly,
Emp. fr. 07.44.
"On the relation of Gorgias to the Eleatic dialectic of Being, cf W. Bricker, Hermes
RG (195K) 425-440.
2. Spurious and Genuine in the Philolaus Fragments
There was no agreement as to what attributes it had, whether it
existed at all, or whether it was expressible. But Philolaus pushes this
controversy aside, entrusting its solution to a divine intelligence.*°
He is clear that there are things that are known to us (ἐόντα καὶ γιγνω-
σκόμενα), and that their origin would have been impossible if—as
Gorgias thought—‘‘nothing”’ existed. We cannot go beyond this;
nevertheless the world, as it is given us, can be understood, as a
harmony of opposites.®°
Like the question of being, that of the origin of harmony and its
“‘arrival”’ is avoided, with the explicit phrase ᾧτινιῶν ἄδε τρόπῳ ἐγένετο.
It is simply a necessary prerequisite for the cosmos, just as ἐστώ is
prerequisite to the πράγματα determined by the opposites. Raven
thinks he sees here the clearest proof that the fragments are a post-
Aristotelian forgery: “It is surprising...to find the author...
expressing perplexity about what seems to have been the most im-
portant constituent in his whole cosmology.” Aristotle speaks of the
ἀπορεῖν of the Pythagoreans in a very similar context, in relation to the
formation of the first One. Raven’s argument is that if the Pythag-
oreans had themselves discovered the gap (“omission”) in their
system, they would have filled it up; if it is noticed and left, a
forger mindlessly copied Aristotle.*! The assumption here is that it is a
primary concern of a philosopher to have pat answers to all possible
questions, and a fully consistent system. In fact, this is rather the con-
cern of the doxographer, but in a real effort to get at truth there is
always an awkward “remainder” that will not fit in, or perhaps an
unfilled gap. Scoon aptly mentions that the Νοῦς of Anaxagoras
89 Cf. Xenophanes fr. 36, Alcmaeon fr. 1, Her. frr. 78-79, Democr. fr. 8, Hippoc.
Vict. 1.1, and Epimenides fr. 11. Somewhat more advanced and skeptical is Ecphantus
DK 51.1: μὴ εἶναι ἀληθινὴν τῶν ὄντων λαβεῖν γνῶσιν, ὁρίζειν δὲ ws νομίζειν.
90 It is a basic idea of Heraclitus that ἁρμονία presupposes an opposition (frr. 8, $1; cf.
A2z2). It is also found, later, in the Pythagorean pseudepigrapha: ps.-Archytas p. 20.4
Thesleff: τὰ δ᾽ ἐναντία συναρμογᾶς twos δεῖται καὶ ἑνώσιος (Bywater 33).—Nicom. Ar.
1.6.3; πᾶν δὲ ἡρμοσμένον ἐξ ἐναντίων πάντως ἥρμοσται καὶ ὄντων ye: οὔτε γὰρ τὰ μὴ
ὄντα ἁρμοσθῆναι οἷά τε οὔτε τὰ ὄντα μέν, ὅμοια δὲ ἀλλήλοις, οὔτε τὰ διαφέροντα μέν, ἄλογα
δὲ πρὸς ἄλληλα. This amounts to a kind of paraphrase of Philolaus 6.—Syrian. Met.
165.33f asserts that Philolaus taught that τὸν θεὸν πέρας καὶ ἀπειρίαν ὑποστῆσαι, which
assumes a highest principle over and above the pair of opposites (followed by Boeckh,
53f, 148). Zeller (I 480 n. 1) finds in Pl. Phib. 23c corroboration for this, although Proclus
(Theol. Pl. 3.7 p. 132) makes it clear that only Limit and Unlimitedness go back to
Philolaus, whereas God as a highest principle is Platonic. The One above the pair of
opposites, which Eudorus ascribes to the Pythagorcans (above, ch. I 3, n. 45) is so impor-
tant in Neoplatonism (cf. Dodds, CQ 1928; Procl. In Tim. 1 176.9f) that this in itself
rouses suspicion of Syrianus’ statement. Philolaus fr. 8, ἕν ἀρχὰ πάντων, need not be
understood in this sense; it can be interpreted as referring to the πρᾶτον ἁρμοσθέν (fr. 7).
"Raven, PyHl o8f, Raven in KR 310, Arist. Met. ro8ob20 (ch togtat 2th): ὅπως δὲ
τὸ πρῶτον ὃν συνέστη ἔχον pdyeos, ἀπορεῖν ἐοίκασιν.
ee ee ee αω αν. νῶν σνῶ
presents an analogous problem." According to the commentator of
" | jek ee (col. 17), Zeus, the divine Aér, brought ἁρμονία.
tilolaus avoids mentioning the divine. How harmony came to the
world is passed over just like the problem of ἐστώ; the world is th
and it is in order. | veneers
The question of ἄπειρον and πέρας was much canvassed in Greek
philosophy. Anaximander takes the ἄπειρον as his starting point
Parmenides declared that “Being” was perfect and ἐπ χω limit 4
(fr. 8.42ff), Zeno dealt with the antinomy of πεπερασμένον and ἄπει :
under the assumption of the existence of plurality (fr. 3), and Μεϊίοις
decided that being is ἄπειρον (frr. 2-4). Philolaus takes : stand on the
question, saying that περαίνοντα and ἄπειρα exist side by side, brought
together by the agency of harmony. As others, faced with che maul
plicity of the world, picked certain leading ideas as central—findin
the basic outline of reality in pairs of opposites like warm and asl
wet and dry, or like Parmenides analyzing the whole world of a :
pearance as the mutual influence of light and night—in the same es
Philolaus takes the one feature of limitation as having d ned
the nature of the cosmos. ane
But how can one speak of a plurality of ἀπειραῦ The air can Ὁ
called unlimited, or space, or Melissus’ Being,® but none of th :
has a plural. One can speak of some things as “unlimited in sande!
(ἄπειρα τὸ πλῆθος),54 but περαίνοντα is not the opposite of this. Now
Anaxagoras, following Zeno’s lead, spoke of an ἄπειρον of he smal]
of a never ending process of division.® Divisibility, dependent on the
empty space in the middle,” plays a role in the Pythagorean equation
of even number with unlimitedness.% If the ἄπειρα mpd ar
thought of in the context of endless divisibility, the plural is ae
82 Scoon, GrPh 143f.
98 Anaxagoras fr. 1, Melissus fr. 6.
*4 Anaxagoras fr. 1, Zeno fr. 3.
95 A
ae ΠΣ le frr. I and 3 es tae the idea of the infinitely small; Zeno fr. 3 (cf.
: ὄντα ἄπειρα Which result from the assumpti si
ec Ana tua aes mption of plurality, polemically,
% Abov
Pac Abe ch. 1 2, ᾿ 31..--λτηήβιοες argues, at Phys. 204a8ff, against the idea of the
ὅτ as g οὐσία, An οὐσία as such cannot be divisible into a plurality of infinities
1 ἄπειρα εἶναι τὸ αὐτὸ ἀδύνατον (204a25f), and yet it is ἃ ποσόν. ὥστε ἀτόπ ι
ν ἀποφαίνοιντο οἱ λέγοντες οὕτως ὥσπερ οἱ Πυθαγόρειοί φασιν' ἅ ὑδιδ οἱ 1
τὸ ἄπειρον καὶ μερίζουσι (204432). In this c ei A ceca peice
τ : onnection, Aristotle associates ἄπει i
this con ov with
ἄρτιον (20443 1). Ross relates Aristotle’s criticism to the Pyth eee
Dea Gd a the Pythagorean identification of
oe ᾿ : .e. divisible number, but the criticism is more telling, couched in
rms, if the Pythagoreans actually spoke of a plurality of de.pa—as Philolaus
did.
258
hensible; the opposite, the περαῖνον, is, then-—an ἄτομον. Then the pair
περαίνοντα and ἄπειρα, both of which are called ἐόντα, would correspond
to the Leucippan δέν and pndév-—material atoms and the “empty”
interstices, which do yet “exist.” This close relationship between the
Pythagorean and the atomists, no matter how surprising it seems from
the point of view of the Platonic interpretation of Pythagoreanism,
is actually corroborated by both internal and external evidence. It 15
not only that Leucippus, like Philolaus, belongs in the Eleatic tradition
and scarcely can be outdone by the Pythagoreans’ number theory
in “forcible abstraction.” ®” Aristotle remarked that the atomists, too,
in a certain way made things out to be numbers. The “void” of the
atomists is not a single, endless space, but the plurality of interstices
which make divisibility and plurality possible.°° The cosmos grows
by taking in material from outside.®? It is obviously the Pythagoreans
who inspired the atomists to see in the motes in a sunbeam an indication
of the nature of the soul-atoms.!°° We even have the testimony of a
contemporary that Democritus studied with a Pythagorean; thus
Philolaus and Leucippus are thrown close together} To be sure,
Philolaus maintains a distance from the atomists; he does not speculate
further about being, but looks for relationships in our given, familiar
world and finds them in the ordering function of number.'”
Before that, though, he seeks to prove his basic doctrine; and here
Frank thought he had found an infallible sign of post-Platonic forgery.“
But the refutation of this was already at hand; Reinhardt had shown that
8? Von Fritz, Philosophie und sprachlicher Ausdruck (New York, 1938) 15; Arist. Cael.
303a8; above, ch. I 2, n. 70.
98 ΚῊ 408.
99 D.L. 9.32 = Leucippus At; above, ch. I 2, ἢ. 51; Democr. Ago.
100 Arist, De an. 404a1 et seq.; above, ch. I 3, n. 130.
101 D.L. 9.38: πάντως μέντοι τῶν Πυθαγορικῶν twos ἀκοῦσαί φησιν αὐτὸν Γλαῦκος ὁ
«Ρηγῖνος. Duris(FGrHist 76F23) said his teacher was Pythagoras’ son Arimnestus, Apollo-
dorus of Cyzicus (above, ch. ΠῚ 1, n. 51) mentioned Philolaus. For Leucippus and Philolaus
as pupils of Pythagoras, sce Iam. VP 104. On Democritus’ Πυθαγόρης sec above, ch. II 6,
n. 26; on the Pythagores coins of Abdera, above, ch. II 2, n. 2. Frank argued for the
dependence of the Pythagorean philosophy on Democritus; but the relationship may be
a whole generation earlier. Surely there was an atomist influence on some later Pythagore-
ans. (On Ecphantus, see above ch. 12, ἢ. 74; W. Kranz in Convivium: Festschr. K. Ziegler
(Stuttgart, 1954] 26ff. There could be a relationship between Ecphantus and the cosmology
of the papyrus from Derveni; see Burkert, AGA 1968, 99.)
102 On the other hand, Democritus denied that there could ever be φύσιν μίαν ἐξ ἐκείνων
(sc. τῶν ἀρχῶν), κομιδῇ γὰρ εὔηθες εἶναι τὸ δύο ἢ τὰ πλέονα γενέσθαι ἄν ποτε ἕν. Arist.
fr. 208 = Democritus A37.
103 304f: “... eim ganz entwickeltes dialektisches Bewusstsein, wie es vor Plato...
kaum méglich ist.”
259
what Philolaus presents is a imitation of an Eleatic proofs! first the
possible cases are enumerated, then one after another 1s siminated
until the actual one remains. The method of the ie outraryato Ger
inference, too—" this would not be possible, unless . . .”’—was known
in the fifth century. The author of the book On the Nature of Man
in particular, assures himself of the basis of his anthropology with :
three-stage proof. The preconditions of all genesis are, first, ἃ
plurality of ἐόντα, secondly, their homogeneity, and thirdly, that hey
are mixed in the correct proportions.!% Philolaus’ conclusion as to the
preconditions of all development is similar, though more abstract:
being in general, then the antithesis of limiting and unlimited, and
finally harmony, which is a numerical relationship.
Philolaus also has an epistemological argument: ἀρχὰν yap οὐδὲ
ae yu@anupevon ἐσσεῖται πάντων ἀπείρων ἐόντων (Β3).197 καὶ πάντα γα
μον τὰ γιγνωσκόμενα ἀριθμὸν ἔχοντι' οὐ γὰρ οἷόν τε οὐδὲν οὔτε νοηθῆμεν
οὔτε γνωσθῆμεν ἄνευ τούτου (Β4). Here scholars have thought they
found, if anywhere, the kind of anachronism that proves forgery.
The unknowability of the Unlimited is also Platonic-Aristotelian
doctrine, and number as the basis of all knowledge is emphasized
104 Reinhardt, Parm. 65; cf. Mondolfo, ZM 378. Compare also the “three ways”
of Parmenides, fr. 6. Gorgias, in particular, used the method of elimination. In the oak
given by Sextus (Math. 7.66, 68, DK 82B3), a three-member scheme appears twice
(ἤτοι τὸ ὃν ἔστιν ἣ τὸ μὴ ὃν ἢ καὶ τὸ ὃν ἔστι καὶ τὸ μὴ Ov... ἤτοι ἀΐδιον... ἢ γενητὸν
καὶ ἀίδιον ἅμα καὶ γενητόν). But the threefold division does not appear in ps ~Arist
MXG, which W. Brécker believes to be more reliable than Sextus’ (Hermes 86 [1958]
425-440, esp. 428f, 433f).—Arist. Cael. 274a30ff: ἀνάγκη δὴ σῶμα πᾶν ἤτοι ἄπειρον
εἶναι ἢ πεπερασμένον (cited by Bywater 52) also lacks the characteristic threefold
division.
105 Diogenes of Apollonia fr, 3: οὐ yap dv... οἷόν τε ἦν οὕτω δεδάσθαι ἄνευ νοήσιος
Sea 2; Anaxagoras fr. 12, and still earlier, Heraclitus fr. 23; Zeno, frr. eae andl
a ye 6-7, use hypothetical propositions). Cf. also the method of proof in
106 Hippoc. Nat. hom. 3 (according to Anon. Lond. XIX, written by Polybus the son-
inlaw of Hippocrates - ie., about 400 B.C.): πρῶτον μὲν οὖν ἀνάγκη τὴν γένεσιν γίνεσθαι
μὴ ἀφ ἑνός. πῶς γὰρ ἕν γ᾽ ἐόν τι γεννήσειεν, εἰ μή τινι μιχθείη; ἔπειτα οὐδ᾽ ἐὰν μὴ ὁμόφυλα
ἐόντα μίσγηται. .. οὐδ᾽ ἂν ταῦτα ἡμῖν συντελέοιτο: καὶ πάλιν, εἰ μὴ τὸ θερμὸν τῷ ψυχρῷ
ἀπ τὸ ξηρὸν τῷ ὑγρῷ μετρίως πρὸς ἄλληλα ἕξει καὶ ἴσως, ... ἡ γένεσις οὐκ ἂν yvorro
τὸ γνώδουμενον must be passive (differently from Pl. Crat. 4400 where τὸ
γνωσούμενον and τὸ γνωσθησόμενον are paired against one another). The passive use of
the future middle is common, though rare in verbs with deponent futures (Kuhner—
Gerth i 1.114ff). Yet Antiphon the Sophist used the expression τὸ ὀψόμενον along with
ὄψις, ὀφθαλμοί, ὀπτήρ (DK 87B7), where it must mean the “‘object of the act of seeing.”
Thus the remarkable expression of Philolaus has its parallel in a sth-century author ae
only there.—On γε μήν see Denniston, The Greek Particles (Oxford, 1954”) 3478.
is) Schaarschmidt 66, Bywater 35. The use of γνωσθῆμεν with νοηθῆμεν shows that
νοεῖν no longer means “grasp, recognize,” but “think,” as in Democr. fr. 143. Cf. von
Fritz, CP 40 (1945) 223-242, 41 (1946) 12-34. ΙΗ
260
especially in the Epinomis.!!" Raven thinks ita very important finding
that Aristotle gives no hint of this indispensable epistemological
underpinning of the Pythagorean number theory? But in a passage
of Iamblichus derived from Aristotle we read that “everything 15
revealed through numbers,” and Alexander, using Aristotle’s book on
the Pythagorcans, says that in the Pythagorean view, μήτε yap δύνασθαί
τι τῶν ὄντων χωρὶς ἀριθμοῦ εἶναι μήτε γνωρίζεσθαι ὅλως, τοὺς δὲ ἀριθμοὺς
καὶ χωρὶς τῶν ἄλλων γιγνώσκεσθαι. It is clear that the basic idea of the
Platonic “system of derivation” has had an influence on Alexander's
wording, that is, the distinction of gradations of being according to
whether one can be thought of without the other. But this does not
necessarily imply the idea of ‘“‘knowability”’; and Alexander adds
the words μήτε γνωρίζεσθαι as an afterthought. Thus it seems that
Aristotle did not completely ignore the Pythagoreans’ theory of
knowledge, even if, not being much interested in epistemology, he
failed to mention it in the Metaphysics. The question of knowledge,
once again, stems f-om the Eleatics: οὔτε yap ἂν γνοίης TO YE μὴ ἐόν...
οὐ yap φατὸν οὐδὲ νοητὸν ἔστιν ὅπως οὐκ ἔστι. So said Parmenides
(frr. 2.7, 8.8); and from his time on, the problem of knowledge was a
frequent topic of discussion in the fifth century.™ The thought, then,
that knowledge is only possible through number, and that itis not only
ἄπειρα that exist, but that they must be joined together with περαίνοντα
through the agency of a harmony, is far from impossible in this
century.
“Everything that is known has number.” An example of this is the
numerical relationship of the musical intervals: “Τῆς extent of a
harmony is a fourth and a Gfth.... The fourth is one and one third
(3:4), the fifth one and a half (2:3), the octave a doubling (1:2)””
109 Arist. Rhet. 1408b27f: ἄγνωστον τὸ ἄπειρον (in relation to the theory of rhythm,
where Pythagorean influence is a possibility; cf. Pl. Phib. 17d). For a later formulation,
see Simpl. Cael. 608.231: εἰ ἄπειρα ὄντως ἦν, παντελῶς ἣν ἄγνωστα" ἡ γὰρ γνῶσις ὁρίζει
καὶ περατοῖ τὸ γνωσθέν. ἴῃ Philolaus, however, knowledge is not entirely on the side of
Limit, but is rather a matter of the harmonious coexistence of ἄπειρα and mepaivovra
(fr. 6; fr. 11 is different).—In ps.-Pl. Epin. 976d et seq. number is praised as the basis of
all knowledge; reference is made to the τέχναι (977d-€), music (978a), celestial movements
(977a-b), and φύσις in general (977d et seq., Ὦ obscure hints that obviously relate to the
“system of derivation”). It is unlikely that Philolaus fr. 4 was derived from the Epinomis
passage, as Theiler thought (Gnomon 1931, 351f)—the simple and jejune from the nuanced
and stylish!
τὸ Raven, PyEl 99, KR 310f.—Iam. Comm. math. sc. 25; above, ch. I 2, nn. 112. 118.
__ Alex. Met. 40.12 = Arist. fr. 203 (on Arist. Met. 985b27).
111 See Melissus fr. 8 on sense perception, Gorg. fr. 3 on the unknowability of the ὄν.
Anaxagoras (fr. 12) and Diogenes of Apollonia (fr. 8) emphasize that νοῦς or ἀήρ “knows
everything.”
261
Ht PHTLOLAUS
(fr. 6). “Harmony” here (ἁρμονία) has the special musical sense “scale
an octave long,’ but the essence of musical harmony is the same as of
that harmony which holds the world together. Of the directly quoted
fragments the ordering of the cosmos by number is only alluded to in
the sentence about the One in the middle of the sphere of the universe
(fr. 7); but we may infer from Aristotle’s account that there was also
application of numerical and musical relationships to the cosmos.
The Hippocratic writings illustrate how some thought of the
numerical and musical ratios as bearing on the life of man. For example
Regimen defines its basic question as the determination of the right
relation between nourishment and activity:
If indeed . . . it were possible to discover for the constitution of each
individual a due proportion of food to exercise, with no inaccuracy
either of excess or of defect, this would mean, precisely, the key to
health for human beings.1!8 Ρ ys y
66 - 99 . . .
Health, in “precise”? terms, is a numerical ratio; whoever knows the
numbers has found all he needs.
The numerical ratios have a more special role in embryology. The
growth and health of the embryo depend on whether he finds the
right harmony—and this is expressed in musical terms:
If, on changing position, they achieve a correct attunement, which
has three harmonic intervals, the fourth, the fifth, and the octave,
they live and grow. ... But if they do not achieve the attunement,
and the low do not harmonize with the high in the interval of the
fourth, of the fifth, or in the octave, then the failure of one makes
the whole scale of no value. . . 24
The treatise On Seven-Month Babies is more precise still: the same
numbers determine the course of the embryo’s development and the
course of illnesses: θεωρεῖν δὲ χρὴ οὕτως τριάσι τε καὶ τετράσι, ταῖς μὲν
112 See below, ch. V 2.
48 Hippoc. Vict. 1.2; cf. Iam. VP 163, 244 (probably from Aristoxenus; DK 481):
πε: Pythagoreans desiderated συμμετρία πόνων τε καὶ σίτων καὶ ἀναπαύσεως.
*Hippoc. Vict. 1.8. The correction συλλαβήν, δι᾿ οξειῶν, διὰ πασέων, for the MS
reading συλλήβδην διεξιὸν διὰ πασέων, was proposed by J. Bernays (Heraclitea I, Diss.
Bonn, 1848, p. 18; cf. Literé VII, p. Liv) and, independently, Delatte (Mél. Thomas
aa cf. DK 2201). On De victu (Περὶ διαίτης, Regimen) cf. H. Diller, Hermes 87 (1959)
39-50.
262
Spurious and Genuine in the Philolaus Fragments
τριάσι συνημμέναις, δύο δὲ παρὰ δύο διεξευγμέναις.}5 All the odd numbers
are important, and of the even ones, the 14th, 28th, and 42nd days.
οὗτος yap 6 ὅρος τίθεται TH THs ἁρμονίης λόγῳ πρός τινων καὶ ὁ ἀρτιφυής "δ
τε καὶ τέλειος ἀριθμός" δι᾽ ἣν δὲ αἰτίην, μακρότερον ἂν εἴη ἐπὶ τοῦ παρόντος
διεξελθεῖν. From this it appears that this book is based on some specific
source. In what follows, the doctrine is applied to seven-month and
nine-month children; the fact that eight-month babies do not live is
thought to be due to the imperfection inherent in an even number.
Similar doctrine is found in the treatise Περὶ σαρκῶν: τὸ παιδίον ἑπτά-
μηνον [γόνον] γενόμενον λόγῳ γεγένηται καὶ ζῇ καὶ λόγον ἔχει τοιοῦτον καὶ
ἀριθμὸν ἀτρεκέα ἐς τὰς ἑβδομάδας ... ἐννέα δὲ μηνῶν καὶ δέκα ἡμερέων γόνος
γίνεται καὶ ζῇ, καὶ ἔχει τὸν ἀριθμὸν ἀτρεκέα ἐς τὰς ἑβδομάδας 111 The periods
calculated are 280 days (-- 40 Χ7) and 210 days (=30x7). Other
books as well!!® go into the difference of even and odd numbers;
even numbers are the weaker, more likely to bring misfortune.
The beginnings of numerology, including the special role of the odd
numbers, are older than Pythagoras;"° but, when significant numbers
are thought of specifically as λόγος ἁρμονίας and also defined in musical
terms, the Pythagorean theory of the numerical relation of musical
notes must be part of the background. In addition, we find the same
archaic expressions for the “fourth” and the “fifth” as in Philolaus,
namely συλλαβή and δι᾿ ὀξειῶν, as well as his expression ἀριθμὸν ἔχειν.
There seems to be a direct connection here; and the Hippocratic
treatises, by their forms of expression, are clearly dependent on a more
detailed treatment of these matters than their own. Lucian attests that
Philolaus called the tetractys ὑγιείας ἀρχή,129 and this is to be taken quite
115 Hippoc. Septim. 9, ΝῚ 448 L. The MSS have συνεζευγμέναις, but there is obviously
a reference to the musical terms συνημμένων and διεζευγμένων. ‘“‘Continued”’ triads
make up the series 1, 2, 3; 3, 4, 53 5,6 7--- Thus he means the odd numbers I, 3, 5, 7,
etc. Correspondingly the arrangement of “tetrads” gives the series 1, 4, 7; 8, 11, 14;
τς, 18, 21, etc. (Delatte, Mél. Thomas 165f).
116 “ Perfect” (wrongly “even” LSJ); cf. ἀπαρτιλογία, Hdt. 7.29, Pollux 2.120, Harpocr.
s.v. = Lysias fr. 28 Baiter-Sauppe, ἀντὶ τοῦ ἀπηρτισμένος Kai πλήρης ἀριθμὸς.
17 19, VII 612 L.; K. Deichgraber, Hippokrates iiber Entstehung und Aufbau des menschlichen
Kérpers (Leipzig, 1935) 20.
118 Epid. 3.17.10, Il] 132 L.: οἱ δὲ mapotvopol .. . ἐν ἀρτίῃσιν μᾶλλον. Acut. II 250-252 L.
119 Below, ch. VI 4.
120 Philolaus ArI = Lucian Laps. 5. It is not clear, either from the Philolaus testimonia
or from the hints of the Hippocratic books, precisely what calculations they made.
According to late sources, by virtue of the tetractys, from the one tetrad of harmonious
numbers 6, 8, 9, 12, or from the other one 6, 9, 12, 18—-whose sums were, respectively,
35 and 45-—one arrived, by multiplying these figures by the “perfect” number 6, at the
total of 210 days for the partus minor (7 months) and 270 for the partus major (9 months).
(Cf. Varro ap. Cens. 9, ‘Th. ar. $1, 63, Anatolius p. 8 Heiberg, Aristid. Quint. 3 p. 142,
Macrob. Som. Sc. 1.6.15ff, Procl. fa Remp. Il 34, ete.) Delatte (Mél. Thomas 171) believes
263
Hh PHILOLAUS
literally: the “tetrad” of harmonic numbers determines growth and
health. So we perceive in the Hippocratic corpus reflections of
Pythagorean doctrines, which were probably in written form; and
the most likely source is the book of Philolaus.
We can learn something from these reflections about the application
and the meaning of Pythagorean number theory. But Philolaus is
concerned with still other relations between numbers and reality:
Now number has two special types, odd (περισσόν) and even (ἀρτιον),
and a third that comes of the mixture of these two, the even-odd
(ἀρτιοπέριττον). And each of these two has many forms, which each
thing individually reveals” (fr. 5)?! It is merely a truism to say that
there are odd and even numbers,!”? but just as the cosmos contains
mepaivovra and ἄπειρα side by side but transcended by a third, harmony,
so here we have an antithesis of two members, and a third above them.
Without any doubt, going by the data of Aristotle and a lexicographical
note in Hesychius,!?* the odd numbers are the περαίνοντα and the even
the ἄπειρα. The ἀρτιοπέριττον, made from the mixture of the two, is
the πρᾶτον ἁρμοσθέν, that is, the One. The basic outline of reality,
that the allusions in the Hippocratic writings prove that this calculation is early Pythag-
orean. But different numbers, which are hard to bring into connection with music, are
given by Empedocles (A83, fr. 153a; cf. A75, fr. 69) and Hippo (A16). Hippoc. Nutr.
42 calculates four possible series for arriving at the duration of pregnancy; and we find
still other constructions in Diocles, Strato (frr. 97-98 W.), and Aristid. Quint. 3, p. 15rf.
—Empedocles spoke of a λόγος μίξεως in relation to the origin of the bones, etc. (A78).
151 Fr. 5. In this sense εἶδος is pre-Platonic; cf. Democr. fr. 11: γνώμης δὲ δύο εἰσὶν ἰδέαι,
ἡ μὲν γνησίη, ἡ δὲ σκοτίη (on the history of the word, Taylor, VarSocr 178-267;
Wilamowitz, Platon II 249ff). On πολλαὶ μορφαί, cf. Eur. fr. 839, Alc. 1159, etc. The terms
περιττόν and ἄρτιον are found as εἴδη τοῦ ἀριθμοῦ also at Divis. Arist. ch. 64 Mutschmann.
The final words are somewhat disquieting; αὐταυτό is Doric, and has the sense of ἑαυτό
(Archytas frr. 2, 3; Epicharm. fr. 172 Kaibel = DK 23B4; Sophron fr. 19 Kaibel; an
inscription from Heraclea, Bechtel, GrD II 403, 255f); but it does not mean αὐτό.
Euripides has a number of expressions like αὐτὸ σημανεῖ (Hel. 151, Andr. 265, Phoen.
623, Bacch. 976).
122 Cf. below ch. VI 4.
123 Above, n. 73; ch. I 2, ἢ. 24.
124 Jn later terminology a number is even-odd if is even and each of its halves is an
odd number (as early as Arist. fr. 47; see Nicom. Ar. 1.9.1; see below, ch. VI 2), so that
“even-odd” is a subdivision of “‘even.” If one understands ἀρτιοπέριττον in this sense in
Philolaus (Newbold 185), his neat classification is spoiled. This inspired Becker (QSt 4
[1936] 1558) and Junge (DtMath 351, C&M 1958, 64) to interpret ἄρτιον as ἀρτιάκις
ἄρτιον μόνον (numbers of the type 2"); but this is unexampled. The only choice left is to
follow the indication of Aristotle (above, ch. I 2, n. 39) and understand ἀρτιοπέριττον
as meaning the One. This fits in with the statement that “‘ both εἴδη have ‘‘ many forms,”
implying that the third does not, and it also comports with the fact that in fr. 7 the One
is characterized as ἁρμοσθέν, which implies that it carries an antithesis within itself. To
be sure, this is unsatisfactory from a mathematical point of view (Becker 159), but not on
that account non-Pythagorean.
264
a, Spurlous and Genuine in the Patiotaus Fragments
suggested in the words περαίνοντα καὶ ἄπειρα, signifies from the begin-
ning that “everything has number.” Things “have even number,
insofar as they contain ἄπειρα; they “have” odd, insofar as they consist
of mepatvovra.2® Both clements are bound together in ‘ harmony,
just as the harmony of music always consists in the connection of even
and odd numbers.’
Aristotle’s usual formulation is that, according to Pythagorean
doctrine, things “are numbers’ or “consist of numbers,’!2? while
Philolaus speaks of them as “having number” (ἀριθμὸν ἔχειν). But
Aristotle also uses this expression in a passage clearly under Pythagorean
influence, at the beginning of the book On the Heavens: “For, as the
Pythagoreans say, the whole and everything in it 1s comprehended in
the number three: for end and middle and beginning [taken together]
have the number of the whole (τὸν ἀριθμὸν ἔχειν τὸν τοῦ παντός), and
this is that of the triad.”’28 This is a remarkable place to find the word
“number.” One would expect something like λόγος, and only after
that, the introduction of the concept of “number.” Beginning, middle,
and end, considered together, “have” or “contain” or “comprise
the idea of a “whole” or “totality,” and since they are three in number
they “‘have” three. A point must be raised here that has been well
made by others:!2° that the Greek word ἀριθμός is not completely
equivalent to the modern concept of “number.” ’ApiOuds is always a
whole number, and tied up with the actual procedure of counting.
Thus it is closely connected with things, and in fact is itself a thing,
or at least an ordering of things. ’Apiuos means a numerically arranged
system, or its parts. Isocrates represents the mythical king Βύθιεις as
establishing the castes of priest, artisan, and warrior in Egypt: anavias
δὲ τοὺς ἀριθμοὺς περιλαβών, ἐξ GE ἀριστ᾽ ἄν τις τὰ κοινὰ διοικήσειεν,
, , 130
ἀεὶ τοῖς αὐτοῖς τὰς αὐτὰς πράξεις μεταχειρίζεσθαι προσέταξεν ...
Ψ κι i
125 Cf. Arist. Phys. 203a10: τὸ ἄπειρον εἶναι τὸ ἄρτιον' τοῦτο γὰρ ἐναπολαμβανόμενον
it ὑπὸ τοῦ ῦ ‘yew τοῖς οὖσι τὴν ἀπειρίαν.
καὶ ὑπὸ τοῦ περιττοῦ περαινόμενον παρέχει ἢ
126 Below, ch. V I.
127 Above, ch. I 2, mn. 15. .
128 Cel, 268a10ff. On the expression ἀριθμὸν ἔχειν cf. also Pl. Parm. 15 3b; above, n. 117;
below, n. 131. 7
129 Stenzel ZG 24ff; Becker, ZwU 21f. The latter renders ἀριθμός as see
Mannigfaltigkeit” (“ordered plurality”), and compares such expressions as “couple,
“dozen,” and “score.” Brunschvicg, Etapes 34, showed that in pre-scientific thought num-
ber is a thing; and Mondolfo, Inf. 200, emphasizes that the converse—things are num-
bers—is but a short step from this. ee oo ᾿
130 soc, Bus. 16. Cf. Hippoc. Acut. Π 228 L., where τοὺς ὃ ἀριθμοὺς ἑκάστου τῶν
νοσημάτων... φράζειν means the subclassification of diseases into their special forms,
describing their πολυσχιδίη or manifold diversity.
265
Hf, PHILOLAUS
Busiris took “‘all numbers,” that is all the classes, in their state as
numbered and ordered groups that would be useful in the government
of society. ‘ApvBuds stands for that which is counted. A sentence in the
Hippocratic treatise On Generation is to be understood in the same
sense: Injured or lame parents may have healthy offspring, ἔχει ya
Tov ἀριθμὸν πάντα τὸ πεπηρωμένον τῷ ὑγιεῖ.81 He is alluding to ‘hie
setae ἰδέαι of the Hippocratic doctrine of humors; if these four
principles are present in full
“The are ἢ μὲ the whole pea i ἜΝ a4 πὸ nina
"The crippl | at is necessary,
in its numerical order, “‘as well as the healthy.” One may also speak
in the same sense, of τὰ μέρεα ἔχειν πάντα 382 =
A significant aspect is the “‘aristocratic” sound of the word ἀριθμός
It is only what is important that “‘counts;” only the fully ae
effective warrior is ἐναρίθμιος, per’ ἀνδρῶν ἀριθμῷ,138 in contrast to che
δῆμος ἀπείρων (Il. 24.776). To ask about the “number” means to ask
about the essential. Each of the pre-Socratic philosophers, in bold
simplification, selects a certain phenomenon or a single aspect of the
world’s multifarious reality—water, air, fire, the warm and the cold—
and each thinks he has got hold of the one most significant thing
The categories of “having” and “being” are not yet strictly oud
The “is” is not taken as indicating a logical classification, but as ;
statement of what is essential. The book On the Nature of Man develops
the doctrine of four humors with reference to the views of ae
who saw the essence of life in one of the humors—blood or phlegm or
bile. And the theory is expressed in the form, “Man is eet ” (or
phlegm, or bile); they proceed from the observation that at the time of
death the body loses its blood, or phlegm, or bile. “Each man is, th
thought, that which they saw him lose as he died.’34 On the a
hand, one may say that the soul “has” a mixture of fire and water 186
and. this means nothing else than that the soul “is” this mixture )
In a similar way, Philolaus tries to get at what is hindarental in
the cosmos by pointing out the antithesis of “Limiting” and ‘‘Un-
limited,” and the presence of number and harmony. “Everything has
131 ἨΙ |
ase seca aria 11, Vil 484 L. Cf. Ov. Met. 7.126: ““perque suos intus numeros
| mp infans, and also the common Latin phrase omnibus numeris absolut
Hippoc. Vict. 1.7. a
183 Tl, 2.202, Od. 11.449, etc., cited b
, Od. 11.449, etc., y Plato at Phlb. 17e; cf. also Eur. fr. $19. and
ene ae ioe to Aegium or Megara (Deinias FGrHist 306F6 = λ Ἶ ἐᾷ τ
oc. Ναί, hom. 6: τοῦτο δὲ ἕκαστον αὐτῶν ἐνό τῆς he rea
τι ᾿Καξαιδομένον εἶδον αὐτὸν ἀποθνήσκοντα. Peres el μὼ 1;
Hippoc. Vict. 1.7: ψυχὴ πυρὸς καὶ ὕδατος σύγκρησιν ἔχουσα.
266
Spurious and Genuine in the Philolaus Fragments
ea
number” means about the same as “everything is, basically, number.”
Aristotle, in the context of his effort to build a systematic ontology,
would naturally choose this kind of formulation; at the same time,
under the influence of the Platonic theory of numbers, the role of
number would naturally be stressed, and ideas of limit, unlimited,
and harmony would recede into the background. Philolaus, however,
explicitly refused to make any pronouncement about ” Being” (ἐστώ)
and is for that very reason more free to follow up the many relation-
ships of numbers—the meaning of ἀριθμὸν ἔχειν.
The correct approach to the Philolaus fragments is not through the
categories of Aristotelian or Platonic thought. To be sure, they agree,
as far as content is concerned, with the accounts of Aristotle; but the
manner of thinking and the style point rather to the pre-Socratic
period, or, to put it more precisely, to that era, in the second half of
the fifth century, of the coexistence of Eleatic dialectic and Ionic
φυσιολογία. Thus the idea of a post-Aristotelian forgery is out of the
question, and the fragments may be regarded as remnants of that
book which the Pythagorean Philolaus wrote not very long before
400 B.C.
The concepts with which he operates—édvra, κόσμος, ἄπειρον,
as well as φύσει καὶ οὐ νόμῳ (fr. 9)—are borrowed from contemporary
philosophy. It is not in this area that one would find the originality of
the Pythagorean who rises to take his part in this discussion, which
has been going on for so long. Most important, the question of Being
is bracketed out, and the effort is to establish relationships, by means
of the ideas of number and harmony. This step could have led from
natural philosophy to natural science and was a necessary step, if one
was to make any progress, in avoiding the Eleatic ἀπορίαι about Being.
But Philolaus clearly did not recognize its significance. If his goal had
been exact science, he would have been investigating the many
facets of the concept of number, and specific problems would have
taken the place of generalizations.’ For Philolaus, philosophical ideas
and specific items of scientific knowledge seem to have been no more
than a means of expressing and illuminating a preexisting picture, of
a world consisting of a pair of basic opposites, informed by harmony
and defined by number. In Aristotle’s exposition the mythical back-
ground becomes clearer: the antithesis of “limiting” and “unlimited,”
136 Cf, above, ch. 1 2.
137 His pupil Archytas seems to have been much concerned with special problems in
mathematics and acoustics.
207
Ill, PHILOLAUS
of “odd” and “even,” is also that of “male” and “female,” and their
conjunction 1s “ marriage" (γάμος). The whole range of observation
and expericnce is compressed, with resolute abbreviation, into a
formula which singles out as essential only something which is not
observable. What is to be shown is not what the world is, but that the
world is “orderly,” and in all its parts and aspects ruled by harmonious
combination—a postulate that is present also in pre-scientific inter-
pretations of the world.
So that everything may fit neatly into the cosmic pattern, the fire
which envelops the world#88 must have as. its counterpart the
“central fire” alluded to in the expressions τὸ πρᾶτον ἁρμοσθέν, τὸ
ἕν, ἐν τῷ μέσῳ τᾶς σφαίρας ἑστία καλεῖται (fr. 7). Whether or not
the related idea of the movement of the earth is possible in the fifth
century 15 a question to be studied in the light of the history of
astronomy.189
Stobaeus has a passage on the structure of the cosmos, in the chapter
he entitles Περὶ σχημάτων, purporting to come from a book called
Βάκχαι.140 It lacks the usual tinge of dialect; and this is as surprising as
its mysterious title. ὁ κόσμος els ἐστιν,131 ἤρξατο δὲ γίγνεσθαι ἄχρι133 τοῦ
μέσου καὶ ἀπὸ τοῦ μέσου εἰς τὸ ἄνω διὰ τῶν αὐτῶν τοῖς κάτω, «καὶ» 48
ἔστι τὰ ἄνω τοῦ μέσου ὑπεναντίως κείμενα τοῖς κάτω. τοῖς γὰρ κάτω τὸ
κατωτάτω μέρος" ἐστὶν ὥσπερ τὸ ἀνωτάτω καὶ τὰ ἄλλα ὡσαύτως"
πρὸς γὰρ τὸ μέσον κατὰ ταὐτά ἐστιν ἐκάτερα, ὅσα μὴ μετενήνεκται. The
cosmos develops from the center out, in each direction equally.
188. A16; cf. above, ch. I 2, nn. 3-4; above, ἢ. 27.
*%° Below, ch. IV 3. Anatolius has a noteworthy comment about the Pythagoreans
(p. 30 = Th. ar. 6.11ff): περὶ τὸ μέσον τῶν τεσσάρων στοιχείων κεῖσθαί τινα ἑναδικὸν
διάπυρον κύβον, οὗ τὴν μεσότητα τῆς θέσεως καὶ Ὅμηρον εἰδέναι λέγοντα. (1. 8.16). The
point of view is geocentric; but Philolaus too identified the ἑστία in the middle with the
ἕν and the cube with γεωμετρικὴ ἁρμονία, ἀπὸ τοῦ κατὰ τὰ τρία διαστήματα ἡρμόσθαι
eee ds Pose Sy = Nicom. Ar. 2.26.2). Was the central fire, the πρᾶτον ἁρμοσθέν,
140 Fr. 17 = Stob. 1.15.7.
Ἢ Arist. fr. 201: τὸν μὲν οὐρανὸν εἶναι ἕνα...
42 μὰ > ? μ Ww
᾿ The ee ἀπό for the difficult ἄχρι, usually accepted since the time of Meineke
"8 ¥ ᾿
rings with it an awkward repetition. Perhaps ἄχρι could be understood in relation to
a primary phase of cosmic development, ‘“‘as far as the middle,” i.e. until the middle is
formed (τὸ πρᾶτον ἁρμοσθέν).
43 «κκαῦ Wachsmuth, ἔστι «γὰρ» Diels.
144, , :
Beye = μέρος Wachsmuth (n.). Diels reads τοῖς γὰρ κατωτάτω τὰ μέσα ἐστὶν
ὥσπερ τ. a., Which does away completely with the reversal of direction (μετενήνεκται).
On μέρος, cf. P] Phd 11262.----- :
: . : 2.—In the next clause the princi al i ὃς
μ pal manuscri
ἀνωτάτω, Diels τὰ ἀνωτάτω. ᾽ pt, F, has Tw
268
3, Spurious and Genuine in tne ἘΠΉΘΙΗ͂ΜΕ Bragrenns
“What is above is that which is over against the middle, from the
point of view of those that are below; for to those below the lowest
part is like the highest, and similarly for the rest; for both (upper and
lower) have the same relationship to the middle, except insofar as
their positions are reversed.” The author is trying to express the idea
of the relativity of “above” and “below” in the world, but keeps
using these terms because he cannot free himself from the idea of an
absolute up and down. Plato dealt with this problem in a much more
sophisticated way, in the Timaeus; and the doxographers record that
“Pythagoras,” Plato, and Aristotle recognize no up and down in the
cosmos.48 The only other laborious exposition like this is in the On
Sevens of the Hippocratic corpus—a_ fifth-century document :!#?
ἴσοι δὲ τὸν ἀριθμὸν ὅμοιοί τε τὴν ἰδέην ot ὑπὸ τῇ γῇ κόσμοι τοῖσιν ὑπὲρ
γῆς... στε τοῖσι κάτω τάδε μὲν τὰ ἄνω κάτω εἶναι, τὰ δὲ κάτω ἄνω.
Clearly the Philolaus fragment is to be classed with this pre-Platonic
passage rather than with the easy mastery of thought and expression
in the Timaeus; and therefore, even if the dialect difference has been
smoothed out in the course of transmission, it may be regarded as
authentic. And incidentally, the idea of the gradual development of the
universe is another feature which would comport ill with post-
Aristotelian forgery.'*®
Another fragment cited by Nicomachus from the book Περὶ φύσεως
also gives the impression of being old:49 κεφαλὰ μὲν νόον, καρδία
δὲ ψυχᾶς καὶ αἰσθήσιος, ὀμφαλὸς δὲ ῥιζώσιος καὶ ἀναφύσιος τοῦ πρώτον,
αἰδοῖον δὲ σπέρματος καταβολᾶς τε καὶ γεννήσιος" ἐγκέφαλος δὲ «σαμαίνει»
τὰν ἀνθρώπου ἀρχάν, καρδία δὲ τὰν ζῷου, ὀμφαλὸς δὲ τὰν φυτοῦ, αἰδοῖον
δὲ τὰν ξυναπάντων- πάντα γὰρ ἀπὸ σπέρματος καὶ θάλλοντι καὶ βλαστάνοντι.
Of course, scholars have attempted to identify portions of this with
145 For ὑπεναντίως with the genitive, cf. Hdt. 3.80, 7.153. τοῖς κάτω is dative of relation,
as in the next clause. In the translation of DK, the construction of the genitive τοῦ μέσου
ἰς not clear: “‘denn was oben liegt von der Mitte aus, verhilt sich zu dem, was unten liegt,
entgegengesetzt.””
146 P], Tim. 62d. Bywater (528) stigmatized the Philolaus book asa copy of the Timacus.
__Aét. 2.10.1.—On διὰ τῶν αὐτῶν, cf. Epicharm. fr. 1, Reinhardt, Parm. 123.
147 Fiebd. 2; on its date, see below, ch. ΠῚ 3, n. 63. The MS reading (according to
Roscher) is. . . τοῖσι κάτω τὰ δὲ μέντοι ἄνω κάτω εἶναι (corr. Boll.).
148 OF the other citations of the Bacchae, fr. 18 (where only the subject heading περὶ
ἡλίου is preserved) could be brought into connection with Arg (below, ch. IV 3), and
fr. τῷ (a mere allusion) with A14 (both are cited by Proclus). Can Βάκχαι be a late,
“romanticizing” substitute for the title Περὶ φύσεως
149 Fr. 13 == Nicom. in Th. ar. 25.17.
209
Hil. PHILOLAUS
Plato’s three parts of the soul, and their localization in the Timaeus,
as well as the Aristotelian hierarchy of θρεπτικόν-γεννητικόν, αἰσθητικόν,
λογικόν. But such an interpretation demands an incredibly stupid
forger’®° with a remarkably profound conception of nature.5! What
are we to make of ψυχὰ καὶ αἴσθησις in the heart? The words “soul’’
and “perception” are obviously not used in their broader signification;
and this very fact suggests a pre-Socratic milieu. Yuy7} means nothing
more than “‘life,”!*? and αἴσθησις goes with it, not as sense perception
through the various specialized sense organs, but as the apprehension
of stimuli, the faculty of being influenced and reacting 1 Critias wrote
that this kind of αἰσθάνεσθαι was closely connected with the ψυχή that is,
to Πές ;155 what is dead is ἀναίσθητον.
Democritus uses metaphors from botany in speaking of the impor-
tance of the navel; the expression σπέρματος καταβολά occurs in the
passage about Philolaus in Menon’s history of medicine (A27); both
Empedocles and Diogenes of Apollonia know the hierarchy of plant-
animal-man ;!5’ and the distinction between man and beast by the
150 At Pl. Tim. 69d-70e, the three parts of the soul are localized in the head, the breast,
and the belly. Bywater (44ff) thought that the Aristotelian tripartition of the soul and its
functions was mechanically grafted onto the scheme of the Timaeus; but this involves
the oddity of assigning the αἴσθησις, since it is the middle member, to the heart. And how
does the ψυχή come to be associated with it there?—Olerud, too, finds Plato’s tripartite
soul in Philolaus (74f). Can anyone equate ψυχὴ καὶ αἴσθησις with θυμός
151 Schaarschmidt 12f. Frank (320ff) tries by arbitrary combinations to establish a
connection between this and ΑἹΖ (ψύχωσις ἐν ἐξάδι- «οἴ above, n. 41). The gradation
seen in fr. 13 is inconsistent with the scheme of derivation in ΑἸ2.
752 Cf. Diogenes of Apollonia fr. 4: ψυχὴ καὶ νόησις (“life and intelligence”), Hebd.
10, 13, Hippoc. Vict. 1.7. Frank (321) translates correctly ‘‘ Leben.” Cf. also Wilamowitz,
Platon Il 91: “das K6rperliche wiegt so stark vor, dass die Einheit des Seelenlebens,
eigentlich die Seele als solche, itberhaupt noch nicht erfasst ist.”
155 If this were the case, the natural thing would be to localize sense perception in the
head. Cf. Arist. fr. 95.
154 On the pre-Socratic use of αἴσθησις, see H. Langerbeck, Adéts ἐπιρρυσμίη (Berlin,
1935) 44ff. It means “sensation [Empfindung],” but is not used as a general term for the
aggregate of the five senses (cf. Democr. fr. 11, Diogenes of Apollonia fr. 5). In non-
philosophical language αἰσθάνεσθαι means “perceive” or “comprehend” (Soph. 4j.
553; cf. Thrasymachus fr. 1), or “perceive” and “feel” (a misfortune, for example,
as at Soph. OT 424). Plants, too (Protagoras ap. Pl. Tht. 167c), and even lifeless things
have αἴσθησις (cf. Hippoc. Morb. VI 386 L., Vet. med. 15). According to Hippoc. Morb.
VI 392£L., heart and diaphragm have nothing to do with thinking; and yetthey αἰσθάνονται
μάλιστα i.e. “feel” [emotions] most poignantly.” The philosophical concept of αἴσθησις
is worked out in Plato’s Theaetetus as something new (1526, τόος, 186d-e).
155 DK 88A23: τὸ αἰσθάνεσθαι τῆς ψυχῆς οἰκειότατον. Cf. Hippo Art.
156 Fr. 148:... πεῖσμα καὶ κλῆμα τῷ γεννωμένῳ καρπῷ καὶ μέλλοντι... . Bywater refers
to Chrysippus (SVF II fr. 806 = Plut. Stoic. rep. 41.1052: the embryo is καθάπερ φυτόν).
157 Empedocles: Sharastani, DK 1 358f, n.; Diogenes of Apollonia: Arg §44 (distinction
with regard to φρονεῖν). Cf. “plant and animal,” Diogenes fr. 2, and on the differentiation
of man and animal, Alcmacon fr. 1a, Archelaus IK 60A4 86, Eur. ‘Tro. 671f. The hier-
archy plant-animal-man-god is known in the east from Sumerian times (vase from
Uruk, ANEP no, $02) to Iranian (J. Ducheme-Guillemin, East and West 11. [1y62] 200),
2. Spurious and Genuine in the Philolaus Fragments
criterion of vods may be derived from Alcmacon. The tendency to think
in terms of parallels and antitheses, which appears here in Philolaus, i
unquestionably ancient. The thought of microcosm and macrocosm 15
also relevant in this context, insofar as the types of ee are
brought into relationship with the organs of the body.** In
correspondences, once more, we find the orderly arrangement an
“harmony” of the cosmos, here divided into four and to that cae:
defined by number.!® Thus this fragment fits into the general point o
view shown in the others, cven though there is nothing in 1t about
Limit and Unlimited. ,
There is perhaps even less of this in the best-attested passage, that
from Menon (A27-28). Here the subjects are heat and cold, the causes
of disease, blood, bile, and phlegm—all treated quite in the manner of
a physician writing in the tradition of natural philosophy. Living
beings, we learn, develop from “the warm, for the semen is ae as
well as the uterus. After birth the infant draws in air and expels it
again “as though it were a loan.” This is how it cools ἐν ἢ is
explained from ψύχειν as φλέγμα is derived from φλέγειν. WwW ether or
not this is dependent on Prodicus, a relation which would at us a
terminus post quem, is matter for controversy." The closest re ae
ship of the passage seems to be, once again, with τὰν ο
Apollonia! The necessarily eclectic nature of Philolaus ἐτῶν 15
especially clear in this instance. To be sure, a relationship to Pyt a
orean cosmology has been alleged here. Just as, according to Aristotle,
the cosmos breathes in the “unlimited” void, as the One, after its
origin, sucks in the Unlimited, so the newly born, warm living
creature breathes in the cool air; here again microcosm and ee
are set parallel.1% Still, in order for the analogy to be really sean a
general subject would have to be embryology, whereas here : | os
a completely formed infant making its first contact with t : air.
Perhaps even to a Pythagorcan it was more important to Pil | a
ledge of a broad range of subjects than to have a perfect system.
158 Olerud 72ff.
7 -ourse ᾿ above, ch.
159 In a fourfold division the power of the tetractys 15 of course at work (above,
1 3; Il 4; above, n. 120).
160 Cf. above, n. 1. : “ΠΝ
) a ἐ ᾿ ; y θερμασίᾳ δέ
161 Diogenes of Apollonia Α28: γεννᾶσθαι μὲν τὰ βρέφη ἄψυχα, ἐν θερι ᾳ
ae : 5 βρέφους τὸ ψυχρὸν εἰς τὸν πνεύμονα
ὅθεν τὸ ἔμφυτον θερμὸν εὐθέως πρυχυθέντος τοῦ βρέφους τὸ ψι ΧΡ ἐν ΗΝ
Thie is ike Philolaus Α27: the living creature arises from the warm
ἐφέλκεσθαι. Vhis is like Philolaus A27: the living
᾿ - ι Με:
rin, and μετὰ τὴν ἔκτεξιν εὐθέως ... ἐπισπᾶται τὸ ἐκτὸς πνεῦμα ψυχρὸν ov .
warlll, ¢ de a av € Os
als saounethy te sitailar,
Hippo, Aro also has something qui | | | _ ᾿
ὦ Above, ch. 1, ἢ,.47. Phe connection with Philolaus A27 was seen by Frank (3271),
Mondolfo in ZM (470), Olerud Ge), and Suaven (Gn KE 304).
Hl PHILOLAUS
Philolaus could draw on a well established Sicilian and south Italian
medical tradition, with names like Democedes, Alcmacon, Empedocles,
and Acron.18
A medical milieu is also the source of the doctrine that the ψυχή,
“soul” or rather “‘life,” is in fact a “harmony” of the bodily func-
tions.!*4 It is ascribed by Macrobius to “Pythagoras and Philolaus”’ 198
this can come from reliable doxographical tradition, but also may be
an inference from the Phaedo, where this doctrine is discussed. Scholars
saw long ago that this is the only point where Echecrates interrupts
the account of Socrates’ last conversation: θαυμαστῶς γάρ μου 6 λόγος
οὗτος ἀντιλαμβάνεται καὶ viv Kal ἀεί, TO ἁρμονίαν τινὰ ἡμῶν εἶναι τὴν
ψυχήν... 186 This shows that, as Plato represents the matter, the
soul-harmony doctrine was important for this Pythagorean from
Phlius, who was a pupil of Philolaus.1® If the order of the universe is
ἁρμονία, then so is that of organic life, and the passage we have cited
from the Hippocratic Regimen (n.114 above) shows how one could
express the idea of “‘life” in musical terms, too. It is another question
how this concept of harmony comports with the old belief in the
immortality and transmigration of the soul. One-has the impression
that Plato, in this passage of the Phaedo, was the first to point out an
embarrassing implication in the idea of the soul as a harmony.168
163 Ch. HI 3. At A27, the thesis is treated as specifically Pythagorean that illness is
caused, among other things, by ὑπερβολαὶ θερμασίας . . . καὶ ἔνδειαι, so that health naturally
is dependent on the correct balance. This idea, however, is not only found in Alcmaeon
(fr. 4) and Hippo (Air), but became a commonplace in Greek medicine (Keus 58ff,
Kramer 366ff). In this passage of Philolaus, however, it is introduced only as an after-
thought (A27).
164 See Wehrli on Dicaearchus fr. 11 (D., as well as Aristoxenus, frr. 118ff kept this
doctrine alive); Heraclitus fr. 67a; Hippoc. Vict. 1.7f. For the soul as a mixture, Par-
menides fr. 16, Zeno Ar §29. The comparison between the soul and music: Pl. Gorg.
482b, 5258, Rep. 443d, 432a, 44Ie, et saep.
#65 Macrob. Som. Sc. 1.14.19 = DK 44A23. The lemma may have read τῶν Πυθαγορείων
τινὲς μέν, ὧν ἐστι PirddAaos (like Aét. 2.30.1 = DK 44A20; Aét. 4.2.3 ascribes to Pythagoras
Xenocrates’ definition of soul [above, ch. I 3, n. 73]). Cf. Plotinus 4.7.8.4 (of ἀμῴφὲ
Πυθαγόραν), Philop. De an. 70.5.
166 P]. Phd. 88d. This was brought out by Déring, AGP 1892, 525ff; cf. also Burnet,
EGP 295f. According to Aristotle, woAAoi . . . τῶν σοφῶν teach that the soul is a harmony
(Pol. 1340b18, De an. 407627). J. Moreau contested the attribution to Philolaus (La
construction de P idéalisme platonicien [Paris, 1939] 374ff), suggesting that the doctrine was
too “materialistic.” The danger is, however, that, coming after Plato, we see Pythago-
reanism through idealistic glasses. |
167 Aristox. fr. 19; above, ch. I 4, n. 40.
168 Cf. also Arist. fr. 45. Wilamowitz, among others (Plafon II 90), believed that the
enlightened Pythagoreans of Phlius no longer believed in immortality. Guthrie (I 310)
emphasizes that the prohibition of suicide (Pl. Phd. 61d-c) presupposes a belief in im-
mortality. An attempt to reconcile this belief with ideas of the soul as a harmony is
made by H. Gomperz, Hermes 1932, 156; Rostagni, Verbo τοις, Carcopino, Bas. 168;
2. Spurious and Genuine in the Philolaus Fragments
There remain several passages in praise of number, cited by Stobacus
in his prologue without indication of the title of the book they are
taken from.!® First comes a pacan on the number 10, then onc on
number in general, which makes possible all our knowledge, is at work
everywhere, and admits no pedSos.!” Throughout there is manifested
an animation that is almost poetic, and a definitely rhetorical confor-
mation of style—in the choice of words like παντελὴς Kat mano PES
σύμφυτον τᾷ TH ἀριθμῶ γενεᾷ, in placement of words (πᾶσι παντᾷ,
οὐδενὶ οὐδέν), in the use of parallelism and chiasmus.!”? The content 1s
simple, sometimes even trivial. Only one sentence, bearing on the
theory of knowledge, is obscure, though not incomprehensible.
Number “in the soul, in harmony with sense perception, makes
everything knowable and mutually agreeable, working like a carpen-
ter’s square, fixing and loosing the proportions of things, each es
itself separately, those that are unlimited and those that are limiting.
Both in content and in expression there appear many poimts of
contact with pseudepigraphic Pythagorean writings, and especially
pseudo-Archytas. Verbal agreement with an expression of Plato's, in
the context of the “nuptial number,” is suspicious; for even though
in this matter Plato is “Pythagorizing,”’ what that means, in his case,
is not copying a striking phrase but following a line of thought. The
author of the fragment uses twice the phrase 7 οὐσία τῶ ἀριθμῶ, three
Wuilleumier $72; and others. One might refer to the expression ager hal
(fr. 6), but there is a danger of importing to the interpretation Brn ον i τ ity a
though Platonic, did not belong to the early Pythagoreans. The idea t Bel . νὴ hike
in proper proportion and the reconciliation of opposites (above, n. 163) is 80 ᾿ ᾿ y
identical with the soul-harmony doctrine, but also not ΟΠ ΓΔ ΟΙΘΕΥ to it, em i a
does signify a threat, and finally destruction, to life. (Déring, AGP 1892 525} found a
contradiction between A27 and A323; contra, Mondolfo in ΖΜ 470.)
169 Stob. I prooem. cor. 3 = frr. 11-12. (In DK I 412.15, Theo Smyrnacus ts erroncously
given as the source of fr. 12.) On what follows, cf. Frank 3 13 n. 1. . — 7
τὸ MSS: ψεῦδος δὲ οὐδαμῶς ἐς ἀριθμὸν ἐπιπνεῖ. Jacobs emendation ἐμπίτνει seems
Oe ἀπορουμένω παντὸς καὶ aa παντὶ..., πολέμιον... τᾷ φύσει τὸ
ῦ ι δ᾽ add ἰκεῖον. .. τᾷ τῶ ἀριθμῶ γενεᾷ. '
ἀν} ἢ rhea ene mean “der Seele gemis,” as Becker interprets (Qs! 4.1 Ὁ" ).
Cf. ps.-Archytas 38.19 Thesleff: ἐν ἁμῖν αὐτοῖς κατὰ ψυχαν ἀκ μὴ τ ΚΕΤΤαΡεν eae
νοῦς, ἐπιστήμη, δόξα, αἴσθησις). From them comes truth, when the four γνώσιες bec
j 39.3 Thesleff. “Agreement with sense perception” becomes clearer in
a Re ἐπικρί gE ὁ νό ἐν λόγον ὅκα μὲν ποτὶ τὸ νοατὸν
ps.-Archytas p. 36.22 Thesleff: ἐπικρίνει δὲ ὁ νόος Tov Λόγον : ue Ai lence
ποτιβάλλων, ὅκα δὲ ποτὶ τὸ αἰσθατόν.... For example, the νόος recognizes the Ἶ oe
the octave as 2:1 by use of the νοατόν, but on the other hand, ὅτι baat it kn ᾽
διὰ τᾶς αἰσθάσιος. This also implies an allusion to the λόγοι in a ὕψει ν μὴ
καὶ ποτάγορα must be predicative adjectives with πάντα, as in Pl. Rep. 540 a τ
προσήγορα καὶ ῥητὰ πρὸς ἄλληλα ἀπέφηναν ; Moderatus (ap. oe fies 32): ak caine
τὰ διαφέροντα... σύμμετρα καὶ προσήγωρα ἀπεργαζομένην. (Here differing ἢ ᾿
273
HT, PHILOLAUS
ams ἁ φύσις τῶ dpuad,!™ and once ἁ τῶ ἀπείρῶ καὶ ἀνοήτω καὶ
ἀλόγω φύσις. Many parallels for all these can be cited from Pythag-
orean pscudepigrapha, and in particular the heaping up of ce
concepts to clarify two opposite φύσεις .174 but both form and content
are uncxampled in pre-Socratic or pre-Platonic philosophy. Here
φύσις 1s not, as in fragment t and fragment 6, an expression Ἢ “all
that exists,’’ and also not the existence, in accord with natural law. of
a particular thing; it is a realm or grade of Being. Precisely this nee
of division or gradation of Being was unknown to the Pythagoreans
according to the testimony of Aristotle; they clung to i sin ΙΕ
perceptible world and did not speak of immaterial οὐσίαι.175 Aone
with this goes the high poetic style of compounds like ΒΚ. Ὁ ei
oe Ἰ : 38 n. 5, 4.15off, who construes : ἁρμόζων αἰσθήσει πάντα (τὰν [“ must be supplied’’]
se ie plea ἀλλάλοις | κατὰ γνώμονας | φύσιν ἀπεργάζεται cwpdrwr'...) The
¥ ie an hic see Newbold 179ff, following Boeckh 142ff; Heath, Math. I 780)
or “‘set-square”’ of the carpenter. In h ics it is. 7
i ase pole οὶ . In mathematics it is the difference
ἢ , more generally, of two similar pol ee
numbers” of this type (above, ch. I 2, n Pa tn een el
, ch. 2, n. 27); cf. Iam. In Nic. 58.109ff. But th
to be used in a much broader wa οὐκ bie eee a νι τινα
y, especially since its etymological signifi
obvious. Iam. In Nic. 18.7 uses it s : Ξ6 ἘΠῚ te Nek
c. 18." ynonymously with κανών. According to Lydus M
Ἢ cas ee 10 is γνώμων by virtue of being πάντα Seeman mien : ᾿ spilovea.
ΣἊΝ ον Ὁ icomachus, ap. Th. ar. 79.23, the τεχνικὸς νοῦς used the number Io "
ῇ Ὡς ᾿ oe ve ἡ δὴν καὶ εὐθυντηρίῳ (cf. Th. ar. 81.15 = Philolaus
πίστις, OTe... δεκάδι καὶ τῆς αὐτῆς μορί λον Grea οὗ
: μορίοις περὶ τῶν ὄν 1
eee [-o: MSS, τοῖς Ast, DeFalco, DK, τῶν aera a βεβαίαν tyoy ἐξ ὑεν
a. ser aes 5 ΤΠ ingen (σωμάτων MS F) is ἀπε ν Διὸς this
: in the passive, and late. Heidel’s idea, that σῶμα = ἀθ
that ecehad = ἀθροίζειν, is without parallel (AJP 1907, 79). Newbold’s eae
pa ee cc δῶν ae to correspond to σχίζων, he proposed συνάπτων;
wv would be possible (cf. H
ee Ρ ( eraclitus fr. 1264), though less probable paleo-
173 Ye cing 5 G
Peis stile beatles γενεά, which has a poetical or mythical ring, and is also
hares : : TO σύμμετρον καὶ καλὸν... Kai πάνθ᾽ ὅσα τῆς γενεᾶς ad ταύτης
174 (Ὁ
Cf. above, ch. I 2, n. 119, esp. ps.-Archytas fr. 1 p. 20.15 Thesleff: τὰν ῥητὰν καὶ
λόγον ἔχοισαν WW ἃ ia
wee, TOY imi
ee x cine i okay καὶ ἄρρητον... There are similar phrases in Aristotle
a , ‘a bf 73. Cf. also Arist. Met. 101043: ἡ τοῦ ἀορίστου φύσις “Burysus”
: : ? 3 ~ “~ :
᾿ 11 Thesleff: δύο φύσιες ἐν τῷ ὅλῳ τῷδε... a μὲν ῥητὰ καὶ τεταγμένα και Ady
OV
ἔχουσα... ἃ δ᾽ ἃ ὙΠ ἦν ἢ ;
Νὴ " ἐς a ᾿ nade a ἀπόντος, καὶ ἄλογος καὶ οὐδεμίαν σύνταξιν ἔχουσα, and above
: epos λόγος of “‘ Pythagoras,” Iam. VP 146 = .
wa ¥ 3 ‘ , ‘ ° 4 —s 164.9 Thesleff: Tav a ιθ ῶ οὐ t
ἀίδιον ἔμμεν ἀρχὰν προμαθεστάταν τῶ παντὸς ὠρανῶ καὶ ya το ΘΗ ee eer
ἔτι δὲ καὶ θνητῶν (MS bei " γᾶς καὶ τᾶς μεταξὺ φύσιος,
Εἰ καὶ ΘΡὴ ᾿ εἰων, corr. Festugiére, Rév. 1338 n. 3) καὶ θεῶν καὶ δαιμόνων
nae a ...(Philolaus fr. 11: θείω καὶ odpaviw Biw καὶ ἀνθρωπίνω dpyd ...)
ee perros ah Heat the suspicious nature of the language, but would like ae
ade the difficulty by understanding οὐσία as “‘ P? inti
i wealth” and pointing to tl i
ewes A oo : Ῥ g to the pre-Socratic
ok ἔργα (I qencnides fr. 10, ete.) Cf., however, the wording of Arist. Cael. 298428
er a sen τ ΤῊ ak λεγομένων τὰ μέν ἐστιν οὐσίαι, τὰ δ᾽ ἔργα καὶ πάθη τούτων
r¢ alleged Philolaus, τὰ ἔργα καὶ τὰν οὐσί Da 5 -
ίαν τῶ ἀριθ
175 Above, ch. I 2. iil
274
2, Spurtous and Genuine in the Philotaus Fragments
παντελής, favorites among the imitators of the ‘Fimacus,!76 and the
propensity for formations in τικός, τ΄ which scems to show a direct
connection with Aristotle.!7®
On the other hand, there are clear contradictions with the other
group of fragments, in spite of their having in common the phrase τῶν
g that might
πραγμάτων τῶν τε ἀπείρων καὶ τῶν Tmepouvdvrwy—somethin
very easily have been οορίοα 175 To be sure, fragments 1-7 have an
ἐστὼ τῶν πραγμάτων, but this is general, comprehensive “being”;
number does not have a peculiar mode of being, and Limiting and
Unlimited are not characterized as two separate principles or φύσεις.
Harmony, in those fragments, stands over the pair of opposites which
first made it necessary, while here the “nature and harmony of
number” is set over against the “nature of the unlimited and un-
intelligent and irrational.” “‘Soul,” ψυχή, is here, unequivocally, the
comprehensive notion knowledge takes place “‘in it,” “in agreement
with sense perception,” in the fully developed sense. It is placed in a
relationship of harmonious tension with thought, that 15, with
number.18®
Thus fragment 11 shows so many suspicious features that it cannot
have been composed in the fifth century B.c., but only after Plato and
Aristotle and along with the rest of the Pythagorean pseudepigrapha.'*?
But the status of fragments 1-7 is not affected by the rejection of
fragment 11, for aside from one easily imitated phrase there is no
apparent connection between them, but in fact a definite difference.
There is nothing in the manner of transmission to suggest that they
should be taken together; in Stobaeus, fragments 1112 are introduced
between “‘Timaeus of Locri’” and Aristotle, Platonically interpreted,
on one side, and pseudo-Archytas on the other.1®?
176 Cf. Hippasus, DK 18.11 (above, ἢ. 50): number is παράδειγμα πρῶτον κοσμοποιίας
εις κριτικὸν κοσμουργοῦ θεοῦ ὄργανον.
177 A. Amman, -txos bei Platon, Diss. Fribourg, 1953.
178 Arist. Met. 982a3ff lists these criteria of the highest σοφία: τὸ τὰ χαλεπὰ γνῶναι,
τὸ ἀκριβές, διδασκαλικόν, ἀρχικώτερον, αὑτῆς ἕνεκεν. Cf. Philolaus: γνωμικὰ γὰρ & φύσις ἁ
τῶ ἀριθμῶ καὶ ἡγεμονικὰ καὶ διδασκαλικά.... (γνωμικός is unparalleled; an casy altera-
tion would be to γνωστικός).
179 Br, τι, DK I 412.2f; fr. 6, DK I 409. rf, cf. frr. 1-2.
180 By contrast with fr. 13; above, n. 154.
181 Another possibility would be that someone issued a new,
Philolaus’ book; but in any case there were several books in circula
modernized edition of
tion under his name
(above, n. 11).
182 Theo’s report (106.10) that
could be based on the genuine book, in which he cert
cryphal fragment.
Philolaus dealt with the number 10 ἐν τῷ περὶ φύσεως
ainly discussed 10, or on the apo-
275
Hl. PHILOLAUS ;
ae 4. Fifth-Century Reflections of Pythagorean Philosophy?
ee ee — up τον 12, Plato alludes in the Philebus and which Aristotle ascribes to the Pythag-
ede reer ese eee ἐπε τὶ of the ὅν that the four orcans. This may well be the only written exposition of Pythagorean
heh εν ἀνε ον δ θα ἔα : τὸ oe nor derived number theory before Plato.!*° The book is hardly likely to have been
Risener ae Κὰ ae ἜΝ ᾿ Si ᾿ ‘ olaus took over a smashing success; perhaps it would seem to us almost as odd as the
“EEL” raises doubts1® O ie pedocles,!®4 the mention of a book On Sevens. The author’s own intentions and the borrowed
ur conclusion must be that here once more philosophical terminology do not always fit harmoniously, so that
we have a popularizing e
PoP g enumeration of the elements by someone much seems tedious or awkward, and much unsystematic or “eclectic.”
Only in the science of Archytas (who was the teacher of Eudoxus) and
in the philosophical reinterpretation of Plato did Pythagoreanism
attain to a form in which its real influence could develop.
We shall have to test the result reached here by considering the
musicological and astronomical doctrines of Philolaus in the framework
of the history of those sciences. Here too we shall find that the Philolaus
fragments represent a stage before Archytas, Eudoxus, and Plato, and
that one cannot, without serious qualification, speak of a specifically
Pythagorean science before Philolaus.
familiar with Platonic and Aristotelian teachings.
Although some dubious material and some that is unquestionabl
spurious has been transmitted under the name of Philolaus, there “8
group of fragments—1I-7, 13, and 17—which can only be interpreted
on the basis of pre-Socratic ways of thinking, not those of Plato and
Aristotle and their followers. Here the proof of authenticity drawn, in
an indirect way, from the doxographical tradition, is directly ae
firmed; there have been preserved for us remains of a book composed
by Philolaus in the pre-Platonic period, including both word-for-word
fragments and doxographical reports, which advocates that philos-
sophy of Limit and Unlimited, of number and harmony, to which
183 Wilamowitz, Platon II 91. The principal opponents of the authenticity of thi
fragment have been Howald (64f), Theiler (Gnomon 7 [1931] 351f), and Festu ‘are (REG
1945, 16 n. 4). Sachs (41ff) was able to treat it as genuine because i does Seay tod
with the regular polyhedra (cf. above, ch. I 3, n. 116; the inclusion of the reference i αν
2.6.5, attributed to Pythagoras, as Philolaus Ars is of course a slender reed).— 7 “a
ὁ τᾶς σφαίρας ὁλκάς, πέμπτον---ἰῃς impressive image of the ship is hard to ἊΝ dip ne
there is no meaningful way to connect this with τρόπεως δικὴν in AI7 (Wilamo as
loc. cit.), nor with Arist. Cael, 291a11, to which Cherniss refers (Pres. 186 n. 177) nets
EGP 294 1, compares πόντος τῆς ἀνομοιότητος, Pl. Pol. 273d, which he thinks should be
equated with the ἄπειρον. Richardson, CQ 1926, 116ff adduced the ὑποζώ f PI,
Rep. 616b; cf. also the references collected by Diels, DK I 413 n. The word Sheds ὁ ane
an embarrassment. It is a technical term of the shipping industry, and presu oe
thing ἕλκον (cf. Wilamowitz, loc. cit.: “das versteht hiemand auch ou er aa
abzuringen glaubt”). The noun ὁ ὁλκός, however, which Wilaniowits proposes ae
aes a “covering” i Wie sphere.” (It is only by way of the Tatin ie
lamowitz can make is way from one meaning to the r
568 ] agrees with Wilamowitz [referring to Orph. me 87.3, eae τος nes
καὶ σώματος ὁλκόν---ἀῃ almost incomprehensible expression, where Dieterich felt ee
aa to conjecture ὄγκον; see Wilamowitz, GidH II 516 ἢ. 2]; see also Mondalfo,
Ἵ δυο Timpanaro Cardini (1946, 331ff) wishes to interpret ὁλκός as the seve
rawn in from the sphere (above, ch. I 2, n. 46); but there are no parallels to support thi
either. The only recourse is to understand ὁλκός as an adjective, with Frank re n 2)
and, before him, Mullach (II 6), who conjectured ὁ τᾶς σφαίρας ὁλκὸς «αἰθήρ» ΤΙ
αἰθήρ causes the movement of the heavens and with a συνεφέλκεσθαι causes all cane ᾿
in the air and on earth (Arist. Mete. 34ta2; cf. Th. ar. 81.19ff), Thus we shall rad: καὶ
Sano paloas Olean: aejiaaur. ees
= rite close connection with Empedocles in the theory of the sun, cf. below, ch. FV 3.
; ccording to A16 (first half), it is nota “fifth elemene” but fire which is the Dé
Cf. P. Moraux, REXXIV tiki 1184, ie ων
seat,
3. REFLECTIONS OF PYTHAGOREAN PHILOSOPHY IN THE
FIFTH CENTURY B.C.?
If the earliest book of Pythagorean philosophy was not composed
until the latter part of the fifth century B.c., the gap between it and the
historical Pythagoras is uncomfortably wide. In order to close it, or at
least to narrow it, many attempts have been made to date individual
doctrines, in the realm of science or natural philosophy, in the early
period; in this way Pythagoras himself could be assured of the credit of
providing the “germ,” or at least the initial impulse, for the later
development. Philolaus himself probably thought of his doctrines as
merely an explication of the wisdom of Pythagoras.! If that were the
case, it would affect our interpretation of the Philolaus fragments:
we should expect more genuine Pythagorcan tradition and less
eclecticism. But when the Philolaus fragments are treated as spurious,
the search for chronological clues becomes really crucial. There 15, 1}
fact, not a single piece of direct evidence to be found. The later tradi-
tion, though it uses the name of Pythagoras, is contaminated with
Platonism and must be ruled out of court2 And the older, reliable
186 Aside from Archytas, though he seems to have devoted himself more to specific
problems, and from Ecphantus, who took his own way.
1 Above, ch. Hii, mn. 98; 14, 9. 20.
2 Above, ch. I 3.
277
Hl, PHILOLAUS
evidence has to do with the Pythagoras legend and the acusmata. It
reveals the picture of a shaman-like sage and a βίος lived in accord with
his precepts,’ a life in which philosophy or science, as the Greeks
understood it, does not necessarily have any part at all. The acusmatict,
later on, did not recognize the mathematici as Pythagoreans.
If there is no direct evidence, can indirect testimony be found, for
example reflections of Pythagorean teachings in the works of other
philosophers? They may have taken over Pythagorean material, or
entered into polemic against it. Since the day of Tannery* scholars
have been treading this path, with growing confidence. They attempt
to discover doctrines of Pythagoras from their influence, as an astrono-
mer sometimes infers the existence of a hitherto unknown star from
irregularities in the course of known planets. In this way a tempting
chapter of the history of philosophy may be built; erratic boulders and
unidentifiable gravel coalesce into a comprehensive structure. The
suspected interaction of the Eleatics and Pythagoreans, in particular,
becomes a living dialogue. Parmenides, the apostate Pythagorean, |
sets up his own system in opposition to that of the school; in re-
sponse, the Pythagoreans revise their theories, only to be subjected
to new attacks, by Zeno; this forces them to undertake further
revision ...
This structure, however, rests on a shaky foundation. It is true that
there are many points of contact between what Aristotle or the later
tradition attributes to Pythagoreanism, and pre-Socratic doctrine; but
to conclude immediately that these come from a Pythagorean origin
would be to presuppose what ought first to be proved. It is also possible
in each case that a later Pythagorean like Philolaus, in an eclectic spirit,
borrowed material from others,® or that the later tradition is wrongly
ascribing to Pythagoras material from a foreign source.” Again, what
3 Above, ch. II 3-5.
4 Tannery, HScH 125, on Pythagoras: “en l’absence de documents authentiques directs
... hous ne pouvons deviner ses opinions particuliéres sur le monde que par les traces
qu’elles ont pu laisser dans les écrits de ses contemporains ou des penseurs de la génération
suivante.”’—Cherniss (Pres. 387, 157 ἢ. 68) considers the ‘“‘number atomism’” deduced
from Zeno so certain that he uses it as a basis from which to criticize Aristotle (206;
cf. ch. I 2, n. 66). Raven (KR 236, PyEl passim) considers that their relationship to the
Eleatics provides our only opportunity to learn any detail or to arrive at any chronological
determination for the Pythagoreans; he knows, though, that this method is “hazardously
conjectural.”
* The comparison is made by J. Stenzel, Metaphysik des Altertums (Munich, 1931) 46,
and Gigon, Ursprung 126.
6 Consider, e.g., Philolaus’ theory of the sun (below, ch. IV 3), and the astronomical
theories of Alemacon (below, ch. IV).
’ E.g., the doctrine of the 4 elements (above, ch. 13, n. 113), or the division of the carth
into § zones (below, ch. IV 1).
278
. Fifth-Century Reflections of Pythagorean Philosophy?
appears both in Pythagorean and in other sources may go hoe oa
common source rather than being the result of mutual influence. n
each case, proof of an “irregularity in orbit would be ᾿ onee
importance in the evaluation of the indirect source, proof of a change
of direction or a distortion of the course of thought and the system
which could only be explained on the basis of external influences. =
this is a very difficult thing to prove, even in the case of Plato, an
seems to lead to nothing but further controversy; for the pre-Socratics,
preserved only in sorry fragments, it is practically hopeless.
Unknown quantities keep multiplying, for the nature and character-
istics of Pythagoreanism, whose influence and diffusion one 1s trying
to determine, are far from being clearly understood. In order to .
any kind of start, one has to take something or other as Bee ADEN :
“siven.” Scholars have frequently regarded it as almost self-evident
that the Pythagorean doctrine of Limit, Unlimited, and number one
have existed from the day of Pythagoras in some form or other, whic
in any case was abstract and philosophical; in this way, the only sae
tion is to decide what aspects of it Parmenides and Zeno presuppose.
But this is the very thing—the existence of these doctrines before the
book of Philolaus—which ought to be proven first; and particular
attention should be paid to clarifying the nature ofa philosophy with-
out written documents. Other scholars, whether consciously or not,
proceed on the assumption that Pythagorcanism had certain general
characteristics: it was a powerful religious movement, the source of
all non-Homeric elements of Greck religiosity ;’" or it is the ies
synthesis of religion and science, featuring the cura animarium : ὰ
special concern of philosophy, and celebrating the harmony 0 ~
and cosmos. Or, finally, it was the source of exact science among "
Greeks, responsible for all significant attainments, especially in ue
realm of astronomy and mathematics.” Each of these hypotheses
8 Cf. above, ch. III 2, n. 31, on "Ὄλυμπος, below, ch. VI 2, on even and odd.
, ch.
1 i 1156.
9 Raven proceeds, tacitly, from this pren . " —
10 To ee out this idea consistently, Homer himself must, it is thought, aoe Ey
| ἂν » . ΗΝ ἢ ‘J a] Ἐκ
revised in the light of Pythagorean ideas. The second part of the ee 2 oe τι ᾿
ἶ ' and ; 600; (ΠΗ,
613: Delatte, Litt. 1353; Od. 11.601fF, and also $76
Pythagorean (Od. 11.565-632: } ; Ὁ ge ae a
, : x he Second Nekyia and the “Gate of dreams”
Symb. 369 ἢ. 5, Lux 190); also t ΠΟ ΤῸ οὐ ὧν
" «62: Carcopino Apitres 202 ἢ. 94, 208 n. 128); pet ay Ἢ ee Deets "
᾿ Ἧι : ͵ I : OU ICEIEUE
: tre, L’Odyssée [Paris, 1924-1925] ad loc,).
RPh 1921] 231; Homere, : 4125 20. : ee
Or fe Pythagorean) is likely to be the forger, thinks Carcopino vee 208 ἢ i :
i Ct me Gigon, Ursprang 12: ὅν ον dass mit Xenophanes cin neues Element ᾿ τὶ
wtthe . 7 9 + β ; >: : . Ἢ τς Ἴ
hilosophie auftaucht νος Und da wir wissen, dass Xenophanes ve | oe ye , x i
hat nennen wir dieses Element Pythagoreisch.” 120: “Thema wird mun das Schicks.
‘lee Seele und die Erzichuny der Seele zu ihrem Schicksale.
l2Chs. IV-VE.
279
Hf, PHILOLAUS
contains something that is attractive, and perhaps a measure of truth;
but as long as they are not defined and supported by direct evidence,
they will not serve as the foundation for reconstruction of the system.
An epoch in which a unique development in the history of thought
took place, like the period of something more than a hundred ycars
between Pythagoras and Plato, surely saw inner transformations even
in apparently stable traditions. In the relation of religion and ration-
alism the center of gravity must have shifted. If we take the legend of
Pythagoras, the acusmata, and the accounts of the acusmatici as seriously as
they deserve, we realize that within his school it was only after the day
of Pythagoras that the movement from myth to science took place. We
also run the danger, influenced by Plato’s evaluation, of seeing the
φυσιολογία of Anaxagoras, for example, in too rational and nonreligious
a light, and disregarding mythical and religious forces which were at
work in this arca as well.
It is a reasonable guess that thinkers from Magna Graecia would
show Pythagorean influences; but only meticulous study of the internal
and external evidence can raise this possibility to a probability—to say
nothing of certainty.
Most important of the relationships between Pythagoreans and other
groups are those with the Eleatics. Geographically these were close to
the south Italian centers of the Pythagoreans, and the abstract, im-
material character of their philosophy naturally results in coincidences
with a philosophy of number. In addition, the ancient tradition makes
Parmenides and Zeno Pythagoreans, or at least pupils of Pythagoreans.!3
Even in Xenophanes Tannery saw polemic against a Pythagorean
teaching. Diogenes Laertius paraphrases his idea of god, ὅλον δὲ ὁρᾶν
18 The statement of Sotion about the hero shrine that Parmenides dedicated to his
teacher Ameinias, the son of Diochaetas, may be derived from a genuine inscription
(D.L. 9.21), though it would still be doubtful whether Ameinias was called a Pythagorean.
For Parmenides and Zeno as Pythagoreans, see Anon. Phot. 439235, Strabo 6, p. 252,
Procl. In Parm, 619.4 Cousin (DK 28A4; Proclus cites Nicomachus as his source, but
there is a variant reading ‘“‘Callimachus,” which is taken into the text by Cousin and
Stallbaum, and later accepted by Zeller I 680.1; the Latin translation makes the reading
“Nicomachus” certain: R. Klibansky, Plato Latinus ΠῚ [London, 1953] xxxii; Callim.
822 Pfeiffer, is among the “‘delenda”). Cebes Tab. 2, Iam. VP 267: Parmenides and
Melissus in the catalogue of Pythagoreans; cf. schol. p. 150 Deubner, Iam. VP 166;
cf. ch. H 4, n. 106. Down to imperial times there was a guild of physicians in Elea which
traced its history back to Parmenides. The president had the title φώλαρχος, which is
reminiscent of the Pythagorean ἄντρα (above, ch. II 3, n. 197). Cf. P. Ebner, Rassegna
storica Salernitana 23 (1962) 4ff; M. Gigante, Parola del passato τὸ (1964) 135-137, 4$0-
452; P. Merlan, AGP 48 (1966) 267-276; H. Jucker, MH 25 (1968) 181-185. The feud
between Pythagoreans and Eleatics, which most scholars have believed in since Tannery,
is not hinted at in the ancient sources, as Capparelli emphasizes (1 267ff).
280
. Pfth-Century Reflections of Pythagorean Philosophy?
καὶ ὅλον dxovew,!4 μὴ μέντοι dvarveiy.'® But according to the Pythag-
orcans the cosmos breathes in the Void, or Unlimited." Xenophanes,
Tannery concludes, is rejecting this idea, thus presupposing that it had
been stated. If this were right, an essential motif of Pythagorean
cosmology would be older than Xenophanes, perhaps suggesting, in
the Opposition of κόσμος and πνεῦμα the theory of περας and ἄπειρον 10
general.” | | |
But this conclusion is anything but certain. Xenophanes is not talking
about the cosmos, but about god, and it is a much mooted question
in the interpretation of Xenophanes whether his god is to be equated
with the universe® But even if this question were to be answered in
the affirmative, it would be just as reasonable to suspect an allusion to
Anaximenes as one to Pythagoras;}® in fact it is impossible to prove
that he is voicing criticism of any particular person. ἘΠΕ ae
reacting against the naive, anthropomorphic conception of the gods.
The principal criteria of a living being, along with the ability to see,
hear, and apprehend psychologically (νοεῖν), are breath and ὉΠ
The god, “as a whole,” exercises the former functions, but not the
latter. There is no more reason to suppose the denial that he breathes
is directed polemically against contemporary philosophers than the
denial that he moves about.”
Parmenides has been exploited much more as a source for Pytnag
orean philosophy, though in a different way. Tannery maintaine
that the doxa section of his poem, with the specific statement that :
teachings are “deceptive” (fr. 8.52), was a doxography, gee a ae
point of view, of Pythagorean cosmology.”! Later writers claimed tha
14 This much is preserved as a directly quoted fragment (24).
16 Γ1,. 9.19 = DK 21Al.
oe , ch. 12, ἢ. 46. . .
1 spat HScH 125ff, followed by Burnet, EGP 108; DK I 113.26 n.; Rostagni,
Verbo 25; Rey 134, 207ff; Mondolfo, ΖΜ 31 4ff. etd he pe ie ΠΝ
: . . . . « ς Ἵ
18 The case for a pantheistic interpretation 15 argued by Geller, ae
ff) believed that world and god we
Tannery, HScH 125. Reinhardt (Parm. 116) i
eae, Xenophanes. Aristotle’s comments in Met. g86broff are of course not con
Oe ae EGP 108, refers to Anaximenes fr. 2. The Pythagorean on ara
by Aristotle can be brought into connection with Diogenes of ἀν κάνη ἜΣ ΤΩΣ "
ἶ i im. 5 again .
th Philolaus (above, ch. ΠῚ 2, n. 161). Pl. Tim 33c argue
"ἐν gy This is of course directed against the Homeric tales of the gods ; ieee it goes
without saying that Homer’s gods breathe. Cf, Aesch. Eum. 651, on Zeus: οὐδὲν ἀσθμαίνων
ee For the interpretation of the doxa section as a se τ : sae ae
icati ans: Tanne cH 232ff, Burnet, ἐ: ;
arm. 63. Application to the Pythagoreans: Tannery, cH 2
i ΠΩΣ aft Mondolfo, ΖΜ 326f, Inf. 286ff, Cornford, CQ 1922, 137, PiParm ff.
281
in Parmenides’ particular manner of developing his argument there
could be seen a reflection of Pythagorean mathematics? Then,
finally, Raven sought to explain the Parmenidean predicates of Being
as a polemical expression against the Pythagorean doctrine of oppo-
sites.24
The world of doxa is, in Parmenides’ poem, the result of opposition
and interpenetration of the contrary powers of Fire and Night. The
attempt to bring this into relation with the Pythagorean opposition of
Limit and Unlimited has led to contradictory results.24 There is not a
single detail that can be shown to be exclusively Pythagorean; there
is nothing in Parmenides about number and mathematics, nothing in
the realm of doxa about Limit and Unlimited, and nothing about the
harmony which unites the opposites. Pairs of opposites played a part in
Ionian physical philosophy from the time of Anaximander, and also
in Alcmaeon. Their reduction into a single opposition is a result of
Parmenides’ fundamental thesis; as he comprehends Being as a unity,
the multiplicity of the world turns out to rest on one basic contradic-
tion.”® Thus the doxa section of the poem is for this very reason a
personal achievement of Parmenides—a fact generally accepted since
22 Gomperz, GrD 136; Rey, 189ff; Cornford, P!Parm 29, PrSap 117. Contra: Szab6, AA
1955, 67ff; 1956, 1o9ff; cf. below, ch. VI 1.
*3 Raven, PyEl 21ff, KR 274. Similarly, K.-H. Ilting, 486 9 (1964) 103-131, tried to
show that πεῖρας-ἄπειρον in Parmenides presupposes a Pythagorean doctrine.
4 Tannery, HScH 207 and 235, equated Night with earth (with Arist. Met. 986b34 =
A24) and with Limit, and Light with Unlimited. On the other hand, Burnet (EGP
109, 186f, followed by Rey 208, 273, 372f) equated Night with πνεῦμα and Unlimited, but
Light with form and Limit. (This is in harmony with the placement of Light and Darkness
in the table of opposites, and fits in with the fire doctrine of Hippasus, DK 18.7.) Mondolfo,
ΖΜ 327, Inf. 345ff, assumes a reversal of positions: the Unlimited was originally a dark
πνεῦμα until, after Philolaus, it becomes the fiery aether (περιέχον, A16); Timpanaro
Cardini argues against this (1946, 328ff).
ὅδ The astronomical system of στεφάναι (A37, fr. 12; below, ch. IV 1), carried on
by Plato’s system of σφονδύλοι (Rep. 616c; cf. Morrison, JHS 1955), comes from the
fiery wheels of Anaximander. On the relation of Anaximander and Parmenides, cf.
Frinkel, WF 186ff. Why Pythagoras is thought to be the intermediary is hard to see
(Tannery, HScH 237).—On Ὄλυμπος, above, ch. III 2, ἢ. 31. ᾿Ανάγκη, absolute necessity
conceived of by thought, is a discovery of Parmenides. (It is of course a misleading con-
vention to spell the word with a small alpha at fr. 8.16 and a capital at 8.30 and 10.6.)
The later tradition has no indication that ἀνάγκη is specifically Pythagorean (Tannery,
HiScH 242ff; above, ch. I 3, nn. 143-150. At Emp. fr. 115 *Avayxn decides the fate of the
soul, but in Pi. fr. 133 it is Persephone, and in the Gold Plates, Moira).—There remains
the “right-left theory” of his embryology in fr. 17 (cf. E. Lesky, AbhMainz 398). The
association of right with male and left with female appears in the table of opposites;
but this leads nowhere except into a realm of pre-philosophical musing about analogies
and “order.” See G. E. R. Lloyd, JHS 82 (1962) 56-66. On the astronomical discoveries
that are ascribed to Parmenides, rivaling those of Pythagoras, see ch. IV 1.
** Reinhardt, in particular, has worked out the argument for the origin of the doctrine
of opposites in the thought of Parmenides (Parm. 71ff, 236ff).
282
the day of Reinhardt” "Phere ts nothing to impel us to insert, between
the lonians and the philosopher of Elea, a specifically Pythagorean
doctrine of opposites.?8 | | ;
According to the view of Raven, Parmenides arrives at his pro-
nouncements on the ἐόν as an alternative to the Pythagorcan doctrine
of opposites. The ἐόν is limited, single, indivisible, and see come
into being or perish, while in Pythagoreanism Limit fies 5 νὰ
against the Unlimited, and in a cosmogonic process the One deve ops
into a Many. Raven himself, however, acknowledges that narnia! :
takes his stand, on principle, against any kind of cosmogony, an
this takes the force from Raven’s argument. For there is no way of
showing that Parmenides is aiming at a specific Pythagorean >
To be sure, his preference for πέρας is significant, and to a modern t /
solution of Melissus seems more natural. But this is merely a mate ο
pre-philosophical valuation; the connection of πέρας and νι is
deeply rooted in Greck language and thought. He explicitly ΕΠ}
the notion of condensation and rarefaction advanced by Anaximenes.
This shows that Pythagoreans were not, at least, his only opponents.
In fact, there is nowhere any unequivocal indication that Parmenides
poem presupposes any Pythagorean science or philosophy.
There may be relationships on a different level. The prologue, in
which Parmenides depicts, in the present tense, a journey in a horse-
drawn chariot beyond the great gate, to a meeting with a divinity and
the revelation of truth, is, as has long been recognized, equivalent to a
27 Reinhardt, Parm., passim; cf. Frinkel, DPh 463f, WF 170ff; senate ;
Raven, too (PyE! 37), argues against connecting the doxa section of the poem with Pythag
ism.
oe This hypothesis has been advocated by Schwabl, WS 1953, 63f, 68f, AAHG 9
6) 148; Ilting (above, n. 23). |
re a arf. On the date of the table of opposites, sce above, ch. 1 2, n. 120. rae
and Raven believe in a reformation of the ela sien sarees as : oe €
iti i ford thinks they put forward their “number m.
critique of Parmenides. Corn emcees
- hich carries the opposites 10 itsedr;
Raven thinks of the One as even-odd, w | PP | sa
had denied that plurality can emerge from unity. On the priority of the One as even odd,
bove, ch. I 2, n. §2.
ΝΣ ΓΟ Cf. Reinhardt, Parm. 50; Szab6, AA 1953-1954, 2471}. Sli ne
been the principal opponent of the attribution of Pythagorean aie ᾿ ae
“tatsachhi icht ein cinziges Wort bet
: ff; cf. 66: “tatsaichlich deutet denn auch nic cin
Se ee Aussenstehendes, Nicht-Eleatisches hin’’); cf. N. B. Booth, Phronesis 2
nores the relationship with the Pythagoreans, in his studies
“Parmenides and the World of Seeming,” AJP 70 (1949)
1 and the socio-political realms.
auf etwas Fremdes,
(1957) 93ff. Frankel rightly ig
of Parmenides. E. L. Minar, | |
41~-$5, sees relationships principally in the persona
283
Hh PHILOLAUS
Journey into the next world."! We may compare it with Aristeas or
Epimenides, as well as the shamanistic healing of illness and the journey
of the dead into the other world found in southern Italian sources
especially the katabasis of Pythagoras.*? Parmenides puts his biiouledae
into competition with older wisdom. Light and Night represent the
realms of life and death. The goddess “sends the souls, now from the
visible into the invisible, now back again. °3 This means that the
existence of the soul is antecedent to the cycle of life and death, and
implies a kind of transmigration. Further, we learn of the Jaiuwv,
a \ “a “ ,
πᾶντα yap «ἢ στυγεροῖο τόκου Kat μίξιος ἄρχει.
“Hateful birth and intercourse”—such a phrase cannot be a simple
formulaic allusion to the pains of childbirth;*4 the attitude of ἐπ ρ ἅδε
and revulsion expressed in the word στυγέειν, στυγερός is contrary
to normal feeling and betrays a remarkable degree of alienation from
ordinary ways of thinking.
κλαῦσά τε καὶ κώκυσα ἰδὼν ἀσυνήθεα χῶρον
—this is Empedocles’ reaction, as expressed in the Katharmoi, to
entrance into human life; and a Pythagorean acusma calls our birth a
punishment. Such a “puritanical” attitude to life, which sees our
existence mainly as a burden and a punishment, can scarcely be called
anything but Pythagorean, especially in southern Italy. This with-
Ἵ Cf. Diels, Parm. off (where, at pp. 14f, we already find the word Schamanismus);
Nestle in ZN 727 ἢ. 1; Morrison, JHS 1955; :
? , 45; Cornford, PrS .
above, ch. III 3. rSap 118; Meuli 171f. Cf.
32 Cf. Burkert, Phronesis 1969.
88. Simpl. Phys. 39.19 (DK ad fr. 13); cf. Rohde, Psyche Η 157f = 372f Eng. ed. (though
he takes no account of frr. 12 and Ar); Zeller, too, recognizes here at least a relationship
with the doctrines of metempsychosis (I 722 n. 2). The fact that perishing is named before
coming-to-be shows that death is not an end.
34 Fr, 12.4. Zeller (I 722 ἢ. 2), opposing M. Ritter, who had seen the correct answer,
would like to interpret oruvyepos here as meaning nothing more than the pains of labor;
but the ὠδῖνες, though they may be πικραί (Il. 11.271) are also ἐραταί (Pi. Ol. 6.43).
Sy ie tg στυγερός surely casts its influence over μίξις. Above all, in Greek usage
eath, or Hades, is στυγερός; στυγερὸς τόκος is parallel to Is "At
life is a death... : i a τ
85 Empedocles, fr. 118; Iam. VP 85. On the concept of “puritanism” introduced by
Dodds, see above, ch. IT 3. One might also refer to D.L. 9.22 = Ar: γένεσίν τε ἀνθρώπων
ἐξ ἡλίου πρῶτον γενέσθαι---ἰῇ the text is accurate. Cf. Burnet, EGP 190 n. 4; differently
Zeller 1 718 n. 1. Also see Pl. Symp. 190b, and below, ch. IV 3, on the mythical descent
of human beings from the sun and the moon. The Pythagorean acusma says that the sun
and moon are the “‘Isles of the Blest”’ (above, ch. IV 4, n. 72).
284
“. PifineCentury Reflections of Fytnagorean entiosopnyr
drawal from the world of the senses, and concentration on the solitary
“Is,” has its prototype in yoga-like exercises in concentration, and in
“shamanistic” cestasy."* Parmenides’ doctrine of Being is, to a degree,
a transposition of the theory of immortality to an entirely new plane.
It is not from Pythagorean science or philosophy that this man takes
his departure, who himself initiated a new era in Greek philosophy,
but from that complex of “shamanistic” prophecy, unusual beliefs
about the soul, and puritanical outlook on life to which the story of
Pythagoras and the acusmata lead us.
Some have thought they discovered in Zeno’s polemical argumenta-
tion still more specific references to Pythagorean philosophy and
mathematics. In Plato’s view he was defending Parmenides against
those who made mock of his doctrine of the one Being, by proving
that the assumption of plurality was even more absurd,” and it has
been thought that the partisans of plurality could only be Pythago-
reans.3® The complex of fundamental problems that makes its first
appearance in the paradoxes of Zeno is still live in modern logic and
mathematics,?® and this gives an additional incentive to understand
Zeno’s arguments as fully as possible and to identify his opponents. To
be sure, even the most determined advocate of his relationship to the
36 Cf. Frinkel DPh? 417-420, and above, ch. ἢ 3.
37 Pl. Parm. 128c, not of course a direct quotation from Zeno about his book, but
Plato’s view of the matter.
38 Tannery, HScH 258ff, and Burnet, EGP 314f, who argue that, since the ΔΘ ἘΝ
and Empedocles were later than Zeno, only the Pythagoreans could fit the circum-
stances and that Zeno was not likely to be attacking ordinary common sense, because
that is inexpugnable. Plato, however, speaks of others “making fun” of him (mwpqdeiv),
and that does not require a formal philosophical system.—Tannery’s thesis was developed
further by Cornford, CQ 1923, 7f, PlParm s6ff, who was supported by others, notably
Lee. Cf. Stenzel, Metaphysik des Altertums (Berlin, 1929), 45 f; Cherniss, Pres. 43 n. 165,
95 ἢ. 401; Mondolfo, Inf. 238ff; on the mathematical side, Hasse-Scholz. It was contradicted
by Zeller 1 752 n. 1; Junge, Symb. Joach. 232ff; Heidel, AJP 1945, 21ff; Vlastos, Guomon
1953, 31; Frankel, WF 234 n. τ; G. E. L. Owen, Proc. of the Arist. Soc. 958 (1957-1958)
199-222; D. J. Furley, Two Studies in the Greek Atomists (Princeton, 1967) 44-56; above,
ch. I 2, nn. 66-74. For the specifically mathematica) aspects, van der Waerden, MtAm
1940-1941; doubts: Booth, Phronesis 2 (1957) 1ff, 99ff; mediating: Mau, Inf.—Plato and
Simplicius speak of a single book of Zeno (Heidel, AJP 1945, 22). According to the Suda
(A2) Zeno wrote, along with other books, one with the title Πρὸς τοὺς φιλοσόφους,
and this is taken as a reference to Pythagoreans (Burnet, EGP 312 ἢ. 2; Lee 8; Joly 31f;
cf. Burkert, Hermes 1960, 170). Zeno was regarded as the ancestor of skepticism (PD. L.
9.72; cf. 9.99); one might say that, just as Sextus Empiricus divided philosophy into logic,
physics, and ethics, and wrote books πρὸς λογικούς, πρὸς φυσικούς, πρὸς ἠθικούς (Math.
7-11), so Zeno wrote πρὸς τοὺς φιλοσόφους, but this is an indication of content, which
became a title only in the Suda; it has nothing to do with Pythagorcans.
39 Sec, for example, Bertrand Russell, Mysticism and Logic (London, 1917, repr. 19$0)
Soff.
440 δ λέξι λα
Pythagorcans® admits that his arguments against movement and
against the idea of space are expressed so generally that it is im ossible
to relate them to any historical persons. It is only the ar ie Ν
against plurality that are referred to specifically Pythagorcan eee
a mathematics of infinitesimals or a “number atomism,” Νὰ
In the directly quoted fragments, Zeno formulates the thesis to b
refuted as generally as possible: εἰ πολλά ἐστιν (frr. 1, 3). This is es
preted, “there is a plurality of conerctc things Ν" ae of ee
concrete bodies is a number, or plurality of units.”4! This wa ‘of
understanding the phrase is supported by the fact that some source oH
signate the “plurality,” against which Zeno is speaking, mores ccificall
as a πλῆθος evddwr*; but, thanks to the thoroughness οἱ Simplicius ‘
can see precisely how this tradition came to be, and can show ie
represents a diversion from Zeno’s own thoughts and man f
speaking. ΝΣ
ἴῃ the second book of the Metaphysics, Aristotle discusses alon
with other problems, those of & and ὄν, in the tradition of Pie:
Parmenides. Here he mentions the view that & is identical with the
point and remarks that the latter would be, xara τὸ Ζήνωνος ἀξίω
nothing.’’48 This is a reference to fragment 2, which we oe
Zcno 8 own words, and nothing else. It is employed in the context of ©
Platonic discussion; and the definition of a point as μονὰς θέσιν ἔχουσ
too, belongs to Plato and not to Zeno."4 ee
Eudemus, also following along Platonic lines, develops the antinom
of ἕν and πολλά, which he regards as solved by the Aristotelian distin ‘
tion between δυνάμει and ἐνεργείᾳ. 45 Then, with a φασί, he adds :
dictum of Zeno’s: καὶ Ζήνωνά φασι λέγειν εἴ τις αὐτῷ τὸ ἐν ἀποδοίη ᾿
40 Tee 64.
41 Cornford, PlParm <8:
ἢ 58; cf. Raven, PyEl 71; Tannery, HS
re - f the history of mathematics, below, ch. VI 1 a 3 ΟΝ οὐ μῶν
ΤΟΣ τ εν Εν χορ τι one with comm. pp. 24ff (Lee introduces these
; ; ᾿ mia first, and only later interprets the verball
oe others, Burnet refers to this evidence (EGP 315: also Raven eee ais
te ( ee of plurality as a “number of units” as a criterion a ne ne
yt howe the idea of “number atomism” at 67ff). General skepticism ᾿ pores
‘rinkel, * 213.1; Heidel, AJP 1945, 23; Booth, Phronesj
speaks of “ones,” as does Zeller I 45. 234 HOON, Krronesis 2 (1957) τ (though he
, 749: ““jede Vielzahl eine Anzahl inheiten ”
Burnet, EGP 291 n. 3, found in ; ΖΔ] von Einheiten’’),—
: * J» the word ὄγκος (Zeno A28 = Ari
the “Stadium”’) a Pytha i 28 = Arist. Phys. 239b33ff,
gorean technical term for the “unit-points”
197). But ὄγκος is used for a physical “ ” “points © (followed by Rey
ἔμ; ysical “body” generally; cf. Pl. Th
fim. 31c, Heraclides frr. 118, 120: fr Υ, Ch Fl. int. 1§sa, Parm. 164d
MW. 3¥c, racrachides irr. 118, 120; frequent in Aristotle. It i
Aristotle is using his own terminology here or Zeno’s pee eget are ὦ
* Arist. Met. 1001b7.
“(ἢ above, ch. I 3. The i int—li
un ee » ch. 1 3. gradation point-line—plane—solid i
ens passage, docs not occur in Zeno’s own aes (fr ἣν which Aristotle mentions
Fr. 37a W. = Simpl. Phys. 97.7ff.
286
4 Piph-Century Reflections of Pythagorean Philosophy ?
ποτέ ἐστιν, ἄξειν τὰ ὄντα λέγειν... This is a very loose kind of citation,
and not a reference to Zeno’s book.4? Eudemus proceeds to explain
the sentence, on his own lincs, beginning ἠπόρει δὲ ὡς ouxe .. . and
by saying “apparently,” he shows that what follows is exposition and
not quotation. In the realm of perception, he says, there is no ἕν
because of the plurality of predicates of each thing—an aporia of
Plato’s#®—but a point is not a ἕν for Zeno, but a “nothing.” Here he is
using the same Zenonian argument as Aristotle. Eudemus then follows
this line of thought further, and concludes with the answer to Zeno,
that ἕν is not ἐνεργείᾳ “many,” but is so δυνάμει.
Simplicius states that Zeno’s argument, in his book, goes in just the
opposite direction. There, as Plato also testifies, he refutes plurality
in favor of the Parmenidean unity of Being whereas Eudemus makes
him prove that there is no One. Alexander of Aphrodisias, however,
continues Simplicius, interpreted the argument set out by Eudemus
differently, as a refutation of plurality: if there is no é, there is also no
plurality, for plurality is a group of unities, a πλῆθος évddwv.* In saying
this, Alexander has Eudemus’ own words before his eyes, as 15 proved
by an explicit citation; that is, he is trying to bring Eudemus’ report
into connection with the content of Zeno’s work—it 15 interpretation
of interpretation. These comments have no value as original source
material for the history of philosophy,®° especially since Alexander's
own source, Eudemus, is not concerned at all with the historical Zeno,
but is merely injecting citations of Zeno into his discussion of a Platonic
problem. Finally, then, when Philoponus adds to the argument from
διχοτομία a further argument of “Zeno” against plurality, that there
is no ἕν because of the plurality of predicates, and therefore no plurality
as τὸ πλῆθος ἐξ ἑνάδων, he is dependent on Alexander of Aphrodisias,*
and what little he adds is merely pedantic systematization.
46 Simpl. Phys. 97.12 = DK 29A16.
47 It is correctly classified in DK (29A16) under the heading ‘* Apophthegmatik”’;
the source could be some Sophistic or Academic dialogue (like Plato’s Parmenides).
Diels thinks of A29, too, as perhaps coming from some ancient dialogue.
48 Cf., e.g., Phib. 14d et seq. ,
49 Simpl. Phys. 99.12 = DK 29A21.
50 Zeller (I 749.2) and Lee (26) are able to regard Alexander as right because they forget
his dependence on Eudemus.
51 Philop. Phys. 42.086 (80.23ff is merely repetition). For the διχοτομία, we have the
original wording (frr. 1, 3), in which he speaks of ἕτερον πρὸς ἕτερον and of πολλά,
but never of ἑνάδες, Alexander also interpreted the διχοτομία in his sense (Simpl. Phys.
138.5); and Philoponus takes this over, as refutation of the ἕν, The second argument
against the ἕν, from the plurality of predicates (where even Socrates makes his appearance
as an illustration), cannot be Zeno’s; this is acknowledged by Lee (2786), though he over-
looks the connection with Eudeimus.
287
The idea that Zeno conceived the plurality that he wanted to refute
as a “plurality of units” is a secondary product of an exposition of
Alexander, who combined the historical Zeno with an argument of
Eudemus’ which was not meant as a historical statement at all. Aristotle
and Eudemus, in discussing the cquation of point and ἕν, cited a passage
of Zeno which had not been written for this purpose. So there is no
justification for using this passage as a basis for inferences about a
Pythagorean “number atomism.”
The origin, meaning, and intention of Zeno’s philosophy will
remain controversial, because the problems he raised are of so funda-
mental a nature; therefore we may leave unsettled the question, to
what extent Zeno’s ἀπορίαιδ8 are intended to prove a positive joa
like for example the idea of a continuum. The connection of Zens
and Parmenides is crucial; Plato himself made it clear that Zeno’s
arguments represented the polemical “reverse” to the “obverse” of
Parmenides’ philosophy of Being. Zeno’s target is the naive world-
view of “sound common sense” in general.5¢ An additional considera-
tion here is that any polemic reshapes its own opponent. The same
realization brings recognition of the correct and rejection of the
false; the antithesis is determined by the thesis. If one is rightly
mistrustful of Aristotle even where his aim is “merely”’ to eae the
opinions of others, because interpretation, as well as polemic, will
always introduce some distortion, then it must be quite a hopeless
undertaking to reconstruct an opposition from Zeno’s polemic, where
Be Lee is forced to assume that Zeno spoke of the One in a double sense: both as the
Parmenidean Being, and as the Pythagorean atomic units—an unlikely aad unprovable
complication. Since Zeno’s first concern is to prove formally that Being must ἜΝ size
(fr. 2), he is not engaging in polemic against a ready-made doctrine of atomic unities
(cf. Vlastos, Gnomon 1953, 32).—Sen. Ep. 88.44(DK 29A2 1) may also come from Eudemus
oF sem the ee δυνατῷ which made Zeno out to be a skeptic (above, n 38).
ἀπορίαι Without solution, dialectic pr ibili pendaate
thesis: Isoc. 10.3, Pl. Phdr. 261d, eae Pie) Siete et nea
ὃ: That Zeno Was aiming to show the continuity of space and time, against a Pythag-
orean atomistic” conception, is the belief of Tannery, HScH 258ff: Burnet. EGP
Cherniss, Pres. 157; Mau, Inf. 1sff, and others. Contra: van der Waerden MtAnn 19 ae
1941. According to Aristotle, the Platonists introduced ἄτομοι γραμμαί pieciscl be
they wanted to get away from Zeno’s paradoxes (Phys. 187a1, Lin. ins. o68ar8 [DK
29A22]; cf. Xenocrates, frr. 41-49: van der Waerden, op. cit. 153; Mondolfo, Inf. 2188 )
Rey (192ff) supposes there were two opposed Pythagorean theories Sue ar which
advocated continuity and the other discontinuity of space and time, so that in whatever
direction Zeno aimed his blows, he would hit Pythagoreans. Surel thi
Pythagoromania ad absurdum! pret sons
55 Pl. Parm. 128ab. Cf. W. Kuilmann, “‘ Ze i ides,”
86 (295) ee ee ΠῚ und die Lehre des Parmenides,”’ Hermes
saat Zeller 1 747 ἢ. 1; Frankel, WF 234 Ὁ. I, 221 ἢ. 2, 229 n. 1; Heidel, AJP 1945,
288
“, Piph-Century Reflections of Pythagorean Philosophy?
an opponent ix not even named. It cannot be proved that there
existed a Pythagorean philosophy or science before Parmenides and
Zeno.
The possibility of Pythagorean influences must be considered in
relation to many thinkers of the fifth century. Ancient tradition brings
Alcmaeon,®’ Epicharmus,®® and especially Empedocles®® into connec-
57 On the form of the name, see DK I 495.39. For Alcmaeon as a Pythagorean, D.L.
8.83, lam. VP 104, 267, Philop. De an. 88.11, Sophonias De an. 14.31, schol. Pl. Alc.
121¢e, Simpl. De an. 32.3. Simplicius emphasizes that Aristotle does not call him a Pythag-
orean: at Met. 986a27ff (on the text, above, ch. I 2, n. 6) a relation between Alcmaeon
and the Pythagoreans is seen, in their doctrines of opposites; but this very point sets
them up as different from one another. Brotinus, Leon, and Bathyllus, to whom
Alcmaeon’s book is dedicated, are regarded as Pythagoreans. (On Brotinus, above, ch.
II 2; the catalogue, Iam. VP p. 144.2 D., names Leon, and at 145.10, “‘Bathylaus” as
Pythagoreans.)—Modern writers emphasize sometimes the originality of Alcmaeon
(Heidel, Medicine 43; R. A. Stella, ‘‘Importanza di Alcmeone nella storia del pensiero
greco,” R. Acc. Linc. 336 [1939] 233-287; Guthrie I 341-359), and sometimes his depen-
dence on the Pythagoreans (Rostagni, Verbo 35f; Mondolfo, ΖΜ 62off; Timpanaro
Cardini 118ff, with refs.). Zeller tries to compromise between the two positions (I 596f).—
If, as Favorinus says, Alcmaeon was the first to write περὶ φύσεως (D.L. 8.83 = DK Al,
cf. A2), he is obviously to be dated earlier than Parmenides.
58 Epicharmus as a Pythagorean: D. L. 8.78, lam. VP 266; with the support of apo-
cryphal writings, D.L. 8.7 (cf. 78), Plut. Numa 8 (DK 23B6s), lam. VP 241ff. Ennius,
in his Epicharmus, deals with Pythagorean doctrines of transmigration (DK 23B47-$4).
Rostagni, more than anyone else, used Epicharmus in the attempt to reconstruct early
Pythagoreanism (Verbo 7ff; cf. Mondolfo, ZM 318ff; Zeller had already taken the
opposite position, I 607ff; Timpanaro Cardini does not include Epicharmus in her collec-
tion). Reinhardt (Parm. 118ff) set out the relations to the Eleatics. Epicharmus cited
Xenophanes by name (Arist. Met. 1o10a5 = DK23B15). In considering the fragments in
detail, one is faced at every step with the problem of authenticity. What Alcimus quotes
may be accepted, tentatively, as genuine (III 1, n. 43; cf. M. Gigante “ Epicarmo, Pseudo-
Epicarmo e Platone,” Parola del passato 8 [1953] 161-175). What looks like a theory of
ideas we can be sure, from the testimony of Aristotle, is not Pythagorean (fr. 3, spurious
according to Diels, DK I 193, and according to Schmid I 644 n. 6, comprehensible “only
if Pythagorean”’). It is not, however, fundamentally Platonic (sce Zeller I 608 n. 6);
we can only guess at the comic context from which it may have come. Is it the contrast
between φυᾷ ἀγαθός and ἀγαθόν as an acquirable πρᾶγμαΐ In that case the fragment
might be genuine.—On even and odd numbers in fr. 2, see below, ch. VI 2.—Among
the entirely or partly apocryphal works, the Politeia (DK 238 56--57) was composed,
according to Aristoxenus (fr. 45 = DK 23A10) by Chrysogonus the flute player (a con-
temporary of Alcibiades; cf. Ath. 12.5354). What Clement quotes from it has a Pythagorean
ring, but also shows affinity with Platonism. Fr. 57.1: 6 λόγος ἀνθρώπους κυβερνᾷ κατὰ
τρόπον σῴζει τ᾽ ἀεί, fr. 56.2: ζῶμεν ἀριθμῷ καὶ λογισμῷ: ταῦτα yap σῴζει βροτούς.
Cf. Epin. 976e: θεὸν δ᾽ αὐτὸν (τὸν ἀριθμόν)... δόντα ἡμῖν σῴζειν ἡμᾶς. Sec also the Derveni
papyrus, col. 20. on the ἀριθμός of the winds and seasons. On the date of Epicharmus,
see Schmid I 1.638f (before 488/487, going by Arist. Poet. 144833).
59 On fr. 129, see above, ch. II 3; Alcidamas ap. D.L. 8.56 (above, ch. II 2); Timacus
FGrHist 566F14 = D.L. 8.54; Neanthes FGrHist 84F26 = D.L. 8.55 (quoting the
‘‘Tclauges” letter); Hermippus, D. L. 8.56. A spurious line of Empedocles names Teclauges
as his teacher (fr. 155; cf. Euseb. Praep. evang. 10.14.15, Theodorct 2.23, Suda 5. v. Emped-
ocles); Alcidamas and Timaeus name Pythagoras himself; and the Telauges letter names
Hippasus and Brotinus. Theophrastus mentioned only his relation to Parmenides, but
289
Hl, PHILOLAUS
ion with Pythagoras himself, but also Leucippus and Democritus.®°
Hippasus™ and probably also Hippo"? were called Pythagorcans.
Modern scholars have seen Pythagoreanism in the book On Sevens,®
and in ancient medical writings generally, as well as in Polyclitus,®
eee Ῥ Ὸ Ὁ Ὸ ΘθΘθΘθὉὋῬΘῬΘὋΘὋΘὋΘὃΘῬΎῪΈΥ͵Ύ ᾿᾽ἪὋὉὃὉὃὖὃὁΦψτ:ΣὃἪὃ'ὃὃὉὃὉὃὁὃ ΄Ξ
Simplicius added καὶ ἔτι μᾶλλον τῶν Πυθαγορείων (ζηλωτής) (Phys. 25.19; Theophr
Phys. op. fr. 3, Dox. 477, with Diels’s note; cf. D.L. 8.55).—Empedocles is eoiienies
cited, quite simply, as evidence for Pythagoreanisin: cf. above, ch. III 1, ἢ. 12
60 Above, ch. JII 2, ἢ. 101. )
$? Above, ch. II 5.
δ DK 38. Zeller treated him among the Ionian philosophers (I 333ff), but Aristoxenus
seems to have mentioned him as a Pythagorean (fr. 21, with Wehrli’s comment Ρ. 54;
Tam. VP 267). He is connected with Pythagoreanism by number speculation (along with
his emphasis on the importance of 7, he mentions 10 as consummatio, τέλειον, Cens.
7.2 = Ατό), with Philolaus by the idea of ψύξις διὰ τῆς ἀναπνοῆς (Alo; above, ch. III
2, n. 161), and with Alcmaeon by his “myelogenic” theory of the origin of semen
(A12; cf. Alcemaeon A13, E. Lesky, Abh Mainz 1950, off ). The epitaph (fr. 2)
ν , ~ - -
Innavos τόδε σῆμα, τὸν ἀθανατοῖσι θεοῖσιν
ἶσον ἐποίησεν Μοῖρα καταφθίμενον
is branded spurious by Diels (DK) and Wellmann (RE s.v. Hippon). Hippo was regarded
as ἄθεος (A2, 4, 6, 8), and Mullach (I 82) understood the epigram as atheistic: the gods are
nothing, just as the dead man is. But Clement, who cites the couplet, understands them as
signifying self-deification, and compares Menecrates-Zeus. This puts him in the company
of Empedocles, in the Pythagorean milieu of Magna Graecia. (Cf. also, in the Gold Plates
θεὸς ἐγένου ἐξ ἀνθρώπου, DK 1B18; similarly 20. Hippo’s place of origin is given as
Mctapontum by Cens. 5.2, Rhegium by Sext. Emp. PH 3.30, and Samos by Aristox.
fr. 21, lam. VP 267.) Even people like Empedocles could fall into disrepute as ἀσεβεῖς:
Hippoc. Morb. VI 358f L. )
** Roscher has the credit for calling attention to this remarkable document. whose
date and relation to Pythagoreanism are still matter for dispute. In any case, the book was
written in Ionia, for Magna Graecia is not included in the “ world map” of ch. 11 (Kranz
NGG 1938, 152). Roscher’s date (time of Anaximander) was accepted by Sarton Hist.
I 21sf (but see Intr. 1 97: “Pythagorean treatise”), and Jones, PhMed. off. On the other
hand, Boll (KiSchr 213ff) and Kranz (NGG 1938, 138 ) have shown that the “‘ world map,”
on which Roscher had based his argument, does not prove anything. Kranz separates
the writer of the book from the conception of the world that he reproduces, and assigns
the latter to the time of Anaximenes (but on ἀνταύγεια of the fixed stars see ‘Metrodorus
of Chios, DK 70Ag9 and Philolaus A19). Boll decided for “450 or even later” (224;
80 Rey 426ff, Mondolfo, ΖΜ 2206), and this seems the most likely solution. It is not eae
whether the author thinks of the earth as spherical (cf. Kahn 84f), but the four elements
are present (ch. 10), and the view of the world’s structure corresponds to that of Hippoc
Vict. The figure of 7 planets plays no part, and this provides a terminus ante quem (below,
ch. IV 1), rendering unlikely the attempt to date it around 400 or even later (Wellmann,
(Qu. uw. Stud. z. Gesch. d. Naturw. u. Med. 4.1 [1933] 6-10 [370/350]; Heidel, Maps ΕΝ
[ca. 400; Festugitre, REG 1945, 21.7). Roscher’s dating would make the treatise pre-
Pythagorean, Pythagoreanism was seen in it by Pfeiffer, Sterngl. 30ff; Wellmann, loc.
cit.; Boyancé, REA 1934, 340; and Mondolfo, ΖΜ 323 (cf. also DK I 406f, n.).
"DK go. Cf. Raven, CQ 1951. But the idea of mathematical proportionality is not
exclusively Pythagorean (below, ch. VI 1-2). The ‘many numbers” and the παρὰ μικρόν
of the single fragment of Polyclitus’ Canon go beyond the radical simplifications of
Pythagorean numerology. Vitruvius 3.1.2ff is a better example of Pythagorcanism, but
Raven thinks this is based on a Pythagorizing source dependent on Polyclitus rather
than the other way around (151). i
290
tM fth-Century Reflections of Pythagorean Philosophy ἢ
Damon," Hippodamus,“” ‘Theagenes of Rhegium,"? Iecus of ‘Taren-
tum,°& Menestor,” Xuthus,7? and even in Anaximenes,” Heraclitus,”
and Socrates.”4
We have already mentioned some general aspects of the problems
these conjectures raise. To regard the fact that a philosopher came from
Magna Graccia, or the close connection of myth and reason, the idea
of a “cosmos,” the presence of number symbolism, or the concept of
85 Damon (DK 37) is not directly called a Pythagorean (Frank 2, 161), though Pytho-
clides, who is brought into connection with him (Schol. Pl. Ale. 118c, DK 37Az2) 1s
(and at the same time is made the teacher of Agathocles the teacher of Pindar!). This is
a shaky foundation for the attempt to derive the whole doctrine of the moral effect of
music from Pythagoreanism (Rostagni, ScrMin I 135ff; H. Koller, Die Mimesis in der
Antike [Bern, 1954] 125ff).
66 DK 39. He was made into a Pythagorean by forgeries in Doric (cf. Delatte, Pol.
125ff; Thesleff, Texts 93-102). The geometrical and modernistic character of his city
plans need not be regarded as Pythagorean (cf. below, ch. VI 1).
δ᾽ DK 8. He was the first to apply allegorical interpretation to Homer. It was a hypo-
thesis of Delatte, often accepted but unproved, that he was a Pythagorean and that the
Pythagorizing interpretations of single lines of Homer go back to his time (Litt. 114f;
most of what is Pythagorean in the Homer scholia comes proximately from Porphyry).
68 Below, n. 79.
69 DK 32. He was the first to study the physiology of plants, following Empedocles
in certain points (Theophr. Caus. pl. 1.21.5 = DK 32.5). Of prime importance was the
opposition warm-—cold, which is also significant to Philolaus (A27) and Hippo (DK
38Ar10). He was a Pythagorean from Sybaris according to Iam. VP 267. Cf. W. Capelle,
RhM 104 (1961) 47-69.
τὸ DK 33, Arist. Phys. 216b26. Simplicius calls him Πυθαγορικός, but probably had no
other source than this passage of Aristotle.
71 A. Chiappelli (AGP 1 [1888] 582-594) tried to show that Anaximenes was depen-
dent on Pythagoras, but this is improbable on chronological grounds if nothing clse.
Burnet thought there was influence, but the other way around (EGP 78f, 108; sec above,
n. 19).
72 If Heraclitus’ originality were not so transparent, people would surely make him
into a Pythagorean. He speaks of ἁρμονία in frr. 8, 10, 51, $4, measure and λόγος in the
world process (frr. 30-31), and has ideas very similar to that of σῶμα-σῆμα (Sext. Emp.
PH 3.230, Dodds, Gorg. 300). See also Gigon, Ursprung 198ff. But what we know with
utter certainty is that he hurled abuse at Pythagoras (frr. 40, 81, 129).
73 On the attempt to trace the theory of ideas, via Socrates, back to the Pythagoreans,
see above, ch. I 2, n. 82 (versus the testimony of Aristotle). After Grote, as early as his
History of Greece (vol. ΓΝ [London, 1847] 335), had seen a connection between the burning
of the Socratic phrontisterion in Aristophanes’ Clouds and the catastrophe of the Pythag-
oreans in Croton, Taylor (Var.Socr. 1ff) interpreted the charge of impiety brought against
Socrates as a measure against newly introduced Pythagorean religion, and saw the
phrontisterion as the site of Pythagorean mysteries: not only initiation and the common
meal, but also natural science, astronomy, and mathematics (129ff). Morrison, CQ
1958, 203, maintaining that Aristophanes “15 suggesting that Socrates’ circle... was...a
Pythagorean synedrion,” thinks this is a really Aristophanic invention, contrary to fact.
But the ritual pattern involving meal and initiation may have been familiar to the
Athenians from Athenian life, from the ἑταιρίαι. A few years later, the “imitation”
of mysteries in one of these aroused a scandal. In the Clouds, it 15 Thales who is named as
the great geometer (Nub. £80). As far as the fire is concerned, the words παῖε and φλέγε
are close together (cf Pratinas fr. 1.01 Dichl).-—It is true, though, that Socrates seems to
291
if, PRILULAUS
harmony,” as a clue to the imthience of Pythagoreanism is to pre-
suppose that Pythagoreanism, as the most powerful intellectual and
spiritual movement in southern Italy, produced a mighty synthesis of
religion and science. The danger in this stereotype is obvious from the
single fact that one inescapably certain point—because attested by the
contemporary Glaucus of Rhegium—namely the relationship between
Democritus and certain Pythagoreans, is felt as almost an embarrass-
ment, and in any case gives us no help in the reconstruction of Pythag-
orean philosophy and science.75
The relation of the south Italian medical tradition to Pythagoreanism
is well worth careful thought. If the physician Alcmaeon wrote in
Croton about 500 B.c., when Pythagoreans formed the ruling group
in the city, there must have been some kind of relationship between
them. Empedocles is embedded in the medical tradition, physicians in
Elea thought of themselves as successors of Parmenides,”* both Hippo
and Philolaus deal with medical topics, and the physician Acron was
Philolaus’ predecessor in the written use of the Doric dialect.7?
Shamanistic γοητεία and medical art, “medicine man” and medicine,
eee
have had something to do with Pythagoreans (above, ch. I 4, nn. 39-40). In the conjuring
up of the dead by Socrates in the Birds (15 53ff), E. Cavaignac sees the earliest testimony
for Socrates’ connection with Pythagoreanism (“Pythagore et Socrate” RPh 33 [1959]
240-248); but the comic poet may well have made Socrates’ ἐπιμέλεια ψυχῆς into a
conjuring up of spirits. On Aeschines’ Telauges see ch. II 5.
“ E.g., the occurrence of the word ἁρμόζειν in the treatise Περὶ ἀρχαίης ἰητρικῆς
Ἢ] been seen as ἃ “fragile... link” with Pythagoreanism (Jones, PhMed 44ff, following
H. Wanner, Studien zu Περὶ ἀρχαίης ἰητρικῆς [Diss. Ziirich, 1939, 18, §4]). Diller has
shown (Hermes 80 [1952] 393ff, 397ff) that this book probably was influenced by Plato-
nisin.— Hebd. 2: ἡ δὲ σελήνη μέση οὖσα συναρμόζει αὐτὴ τἄλλα πάντα (namely the region of
the stars, with the sun and the region of earth) was interpreted by Pfeiffer (Sterngl. 110ff,
comparing Xenocrates, Heinze pp. 75f; Plut. De fac. 945c, Quaest. conv. 7452; cf. above.
ch. I]. 4 ἢ. 148) as a reflection of the Pythagorean doctrine of the harmony of the spheres;
but the word μέση docs not have any specific musical significance here (cf. Kranz NGG
1938, 144 ἡ. 2).
* Cf. above, ch. ΠῚ 2, ἢ. ror. In his ethics Democritus emphasizes μετριότης, συμμετρία
and ἁρμονίη, but this is an emphasis common to many Greeks. If Anaxagoras were not
Unquestionably an Ionian, and labeled a materialist by Plato, Rostagni would have been
tble to interpret the juxtaposition of Νοῦς and the formless mass (ὁμοῦ πάντα χρήματα
ἦν) in the light of the alleged Pythagorean dualism of “forma e sostanza” (cf. Verbo 42).
There are also points of contact with Pythagoreanism in the astronomy of Anaxagoras
as well as that of Alcmaecon (the inhabited moon, dark bodies, and the close relation of
sun and moon; cf. below, ch. IV 1, 3). Were Anaximander not indubitably older than
Pythagoras, he could be more “Pythagorean” than anyone else (the perfect circle, the
iuportance of the number 9, justice in the cosinos).
1. Above, n. 13.
™ Above, ch. III 1, ἢ. 25.
292
1. Pifth-Century Reflections of Pythagorean Philosophy ?
do go together, and in Empedocles the two are still combined.”* A
kind of “knowledge” about the arrangement of the cosmos, which is
partly a matter of number symbolism, forms part of this amalgam,
closely bound up with ritual in the medical prescriptions and proce-
dures which are supposed to lead to recovery. In particular, the transi-
tion from the Bios Πυθαγόρειος (or ᾿Ορφικός) to a rationally based
regimen is only a new approach to the same thing; in each case there
is a system of injunctions to abstinence, whose goal is to concentrate,
to enhance, and to control the powers of the individual. This is how
the trainer Iccus of Tarentum became famous, even before Herodicus
of Selymbria;”® and before him the tradition calls Pythagoras the
inventor of a new kind of regimen,®° and sometimes even calls him a
physician.*!
Of course, one can hardly think of Pythagoras as the only, or even
the most important, originator of these trends. The oldest of the
famous physicians of Croton, Democedes, who was active in the courts
of Polycrates and Darius, is likely to have been approximately con-
temporary with Pythagoras. His father, Calliphon, came from Cnidus,
which shows that the connections between Ionia and Croton were not
all due to Pythagoras.§* Empedocles drew many inspirations from
the medical tradition, and in particular from Alcmaeon,® and in turn
*8 As Hippoc. Morb. 1-2 attacks shamanistic charlatans, Vet. med. 20 attacks the
physiologia of Empedocles. Democedes functioned as a seer in the court of Darius (Hdt. —
3.132). Cf. also F. Wehrli, MH 8 (1951) 36ff.
78 DK 25. Also Ael. NA 6.1, VH 11.3, Lucian Hist. conscr. 35 (Wuilleumier 566). The
dating before Herodicus of Selymbria comes from Pl. Prot. 316d: “Ixxos δὲ ὁ Ταραντῖνος
Kal ὃ νῦν ἔτι ὧν οὐδενὸς ἥττων σοφιστὴς ᾿Ηρόδικος 6 Σελυμβριανός.
80 See above, ch. II 4, for Pythagoras as the alleged reformer of the regimen of athletes.
On the bean taboo, Diogenes Antonius (who sometimes used good sources) says, speaking
of Pythagoras (Por. VP 35), αὐτῷ καὶ τὸ σῶμα ὥσπερ ἐπὶ στάθμῃ τὴν αὐτὴν ἕξιν
διεφύλαττεν, οὐ ποτὲ μὲν ὑγιαῖνον, ποτὲ δὲ νοσοῦν, οὐδὲ αὖ ποτὲ μὲν πιαινόμενον καὶ
αὐξανόμενον, ποτὲ δὲ λεπτυνόμενον καὶ ἰσχναινόμενον. Similarly, Aristox. ap. Iam. VP
τοῦ.
81 ΑΕ]. 4.17 (from Aristotle? cf. above, ch. II 3): ἐπιστρεφομένου δὲ τὰς πόλεις αὐτοῦ,
διέρρει λόγος ὅτι Πυθαγόρας ἀφίκετο οὐ διδάξων ἀλλ᾽ ἰατρεύσων. For Pythagoras as ἃ
doctor see Celsus Med., prooem. 7, D.L. 8.12. For Pythagoreans and medicine, Iam. VP
163, 244, 264, Ael. ΝῊ 9.22; and the acusma, Iam. VP 82, τί τὸ σοφώτατον τῶν παρ᾽ ἡμῖν;
ἰατρική.
82 On Democedes, DK 19, and esp. Hdt. 3.125--137. His father Calliphon is connected
with Pythagoras in anecdote (Hermippus ap. Joseph. Ap. 1.164, DK 19.2). Milo, too,
the father-in-law of Democedes, is connected with Pythagoras (above, ch. II 2).—We
know through an inscription of a physician named Somrotidas of Megara Hyblaea,
about 550 B.c. (SEG 14 [1957] $99).
88 Cf. Kranz, Emped. soff; below, ch. V 1, n. $7.
293
ΠῚ. PHIOLAUS
exercised, himself, a strong influence on the medical literature.*4
Philolaus certainly adopted some medical ideas; yet certain of the
Hippocratic writings seem to be in some way dependent on him.®
The book On Sevens, composed in lonia and in its speculations on
microcosm and macrocosm dependent on very ancient ideas, looks
more like a phenomenon parallel to Pythagorcanism.** Though the
Hippocratic Oath has features in common with Pythagoreanism—the
teacher as “father,” the injunction to secrecy, the religious horror of
suicide, regimen valued morc highly than surgery—much of this is
simply to be explained as the result of a common background.®?
The main question is the source, in all these relationships, of the
really scientific elements,’ from Ionic φυσιολογία and Eleatic logic of
Being, or from a hypothetical Pythagorean philosophy of nature.
And isolated coincidences are not enough to demonstrate the existence
of any such system.
Aristotle himself compared Alcmacon’s theory of opposites with
the Pythagoreans’ “table of opposites,” and Parmenides builds the
cosmos, in the realm of doxa, upon the opposition of fire and night.
8¢ Cf. the polemic at Hippoc. Vet. med. 20.
85 Ch. II 2, nn. 113-117.
ὡ Above, n. 63. One is reminded of Pythagoreanism by remarks on the healing power
of music (ch. 30); but the “7 vowels” (ch. 9) were not yet known in Magna Graecia in
the sth century. On the expression ᾿Ολύμπιος κόσμος, see above, ch. III 2, n. 31. In place
of ἄκριτος κόσμος for the outermost heavenly sphere, Roscher suipwesied Ἢ τος
which Pfeiffer (Sterngl. 336) combined with Philolaus A16 (see above ch. III 2 ee ἢ):
but ἄκριτος is guaranteed by the equation of inseparabilis soliditas with ee ayes
in ch. 6 (a gloss of Galen, VIII 637.1 L.; Kranz, NGG 1938, 124 n.). “Axpuros is “endless,”
Parm. fr. 6.7, Critias fr. 19.4. The idea of macrocosm and microcosm unites On Sevens
with Regimen (see esp. Kranz, NGG 1938), and, more remotely, with Philolaus fr. 1 3
An Iranian origin for this concept was suggested by A. Gétze, “ Persische Weisheit Ἢ
griechischem Gewande,” Zs. f. Indol. u. Iran. 2 (192 3) 60-98, 167-177. See Olerud, passim:
J. Duchesne-Guillemin, in Problemi attuali di scienza e di cultura 76 (Rome +966) 2
has withdrawn the objections he offered in HTR 49 (1956) 115--122. Gotze (86fF ) believed
that the Pythagoreans formed an intermediary link between Persia and Greece, as did
Olerud (212; at p. 220, however, he envisages the tradition as passing from Borda to the
Pythagoreans via Cnidus).
a Cf. above, ch. 4, n. 104. The most striking parallel is the prohibition of suicide
but this makes perfectly good sense in the medical tradition; the physician as prescrver
of life may not promote death. Classification of the Oath as “a Pythagorean document”
by Edelstein outruns the evidence.
Ae: Schumacher, Antike Medizin (Berlin, 19632) 46ff, 81ff, traced the scientific basis of
Greek medicine to Pythagoras (the idea of the regularity of nature, the concept of health
as a norm and as harmony, the responsibility of the sick person to himself, the importance
of the daily regimen). This means, of course, simply deciding a priori that what 15 scientific
is Pythagorean. Jones, PhMed. 1ff, thinks that at the least the combination of philosoph
and medicine is attributable to Pythagoras, though the unity of medicine and δ ἀπεεύδονεν,
ideas was present very carly in the context of shamanism, and the only question is, what
Pythagoras contributed that could be called scientific. ᾿
204
.- Efth-Century Reflections of Pythagorean Philosophy?
Did the background of these thinkers include a Pythagorean doctrine
of Limit and Unlimited? In considering this, let us remember that it is
hardly possible to arrange in parallel συστοιχίαι the opposites named by
Alcmacon, which are taken from everyday observation.®® Does not
this fact, in itself, show that his idea is pre-Parmenidean? The “table
of opposites,’ on the other hand, is probably later, and Alcmaeon
takes up ideas about opposites which had been a part of the Ionians’
thought since the time of Anaximander. Hippasus the Pythagorean
appears in the doxographical tradition as a plain monist;°° Hippo
speaks of moist and dry, warm and cold, in the same way as the Ionian
physiologers;®! and even in Philolaus the warm and cold of his physi-
ology do not seem to be integrated into the concept of the Limiting
and Unlimited of his cosmology.* There is nothing to show that this
theory of opposites was already there at the beginning of the fifth
century. If we restrict our attention to the evidence, it seems much more
likely to be a position taken on the question that rose, in reaction to
the work of Parmenides and Melissus, about the limitedness or un-
limitedness of Being. Number symbolism is to some extent older, but
there are no traces of it in Parmenides, and at the most, very slight
traces in Alcmaeon and Empedocles.% Before Philolaus, Hippasus
seems to be the only one to have dealt with music theory.”
89 According to Aristotle Met. 986a22 (and elsewhere), Alcmaeon mentioned white and
black, sweet and bitter, good and bad, large and small. Aétius (5.30.1 = fr. 4) names
moist and dry, cold and warm; but these are so common that we cannot rely on this
report (cf. Pl. Soph. 242d; Dox. 223). These ideas appear also, however, at Ag and Ag.
Timpanaro Cardini also thinks that the table of opposites is later than Alcmacon (120;
cf. above, ch. I 2, n. 120). Mondolfo, ΖΜ 321f, posits an older, more primitive dualism
in Pythagoreanism as the source of Alcmaeon; but why should we need to insert an
intermediary between the Ionians and Alcmacon? Warm and cold, dry and moist are
already there in Anaximander (AQ, I0, II §§6f, 17a, 27, 28); cf. Anaximenes A21, Hera~-
clitus fr. 126, Kahn tooff, 16off.
80 Above, ch. II 5. Burnet (EGP 109) tried to bolster his equation of Parmenides’
Light with Limit by a reference to Hippasus (cf. above, n. 24), but there is no mention
of an opposing principle in Hippasus, and ever since Aristotle he has had his place beside
Heraclitus.
91 DK 38Ar1, cf. Aro. At the same time, the life-giving “moist” is for him the real
ἀρχή.
95 Above, ch. III 2, on A27. According to the evidence of Menon, our bodies “‘ consist
of the warm,” and the cold air is something foreign, borrowed, so that alongside the
idea of harmony a monistic thought seems to find a place.
98 Cf. Alcmaecon Ars. On Empedocles’ doctrine of the elements, below n. 101. On
dppovia Emp. fr. 27.3, 96.4, 23.4. Empedocles speaks, in fr. 96, of a numerically deter-
mined proportion in the mixture of the elements, and Mondolfo (ΖΜ) decides that this
is an indication of Pythagorean influence. Aristotle, however (Met. 1092b8ff), differenti-
ates between number as the λόγος μίξεως, and the Pythagorean and Platonic concept of
number (Zeller I 1026).
84 Cf. ch. V, below.
295
he PRILOLAUS
Alemacon had thingy to say about astrononry, and in particular the
“contrary movement" of the planets not atopic in which a physician
would obviously be interested. He also tried to prove the immortality
of the soul by its relation to the stars: both were, he thought, in eternal
motion. This has so “Pythagorean” a ring that scholars have often
suspected a post-Platonic forgery." Still, the astronomical views can
be derived from the Ionians,"® and the proof of immortality, which is
not identical with that in Plato’s Phaedrus, seems to be Alcmaeon’s
original contribution.* If this is so, what Alemacon did was to trans-
late pre-scientific, and Pythagorean, material into scientific language;
nothing is presupposed that is Pythagorean except the theory of
metempsychosis and a connection of immortality with the heavenly
bodies, which appears in the acusmata.%®
Empedocles has a perennial interest because of his intermixture of
rational explanation of natural phenomena and religious pronounce-
ments like those of a prophet, both emanating from an eloquent and
passionate personality. We cannot maintain a neat correlation of the
two sides of his nature with his two poems, On Nature and Purifica-
tions;*® and it has been rightly emphasized that this very fact shows
95 Wilamowitz, Platon I 461, 717; Frank vii; Moreau, Ame 155 ἢ. 6. The difficult
point about this thesis is that it requires us to suppose Aristotle let himself be deceived
by a post-Platonic forgery.
96 Below, ch. IV 2.
* Arist. De an. 405229 = Alemaeon ΑΖ, cf. ΑΙ, as well as the expression in fr. 2,
“People perish because they cannot join the beginning with the end” (with an allusion,
no doubt, to formulae of the mysteries about beginning and end; cf. Pi. fr. 1 37a, Her.
fr. 103, Hippoc. Vict. 1.19, Nutr. 9, 24, Loc. hom. 1, Ocellus 1.14). The equation of ψυχή
and movement makes the foundation for the proof of immortality: when life ceases,
the creature stops moving. That which is a conclusion from analogy in Alcmaeon
becomes in Plato, through the idea of self-movement, a dialectical proof (Phdr. 2450).
δ Cf. above, ch. II 4, and below, ch. IV 4.— Whether Alcmaeon taught transmigration
is as controversial as the relationship between the doctrine of immortality and his phys-
iology. (Rostagni, Verbo 1o2ff, distinguishes between an “anima sopranaturale” and a
“natura corporea,” which were confused, through a “grosso equivoco”: cf. Kerényi
28ff, Stella 278ff [above, n. 57], Guthrie I 355f).
" This is emphasized by Long 46ff. A development “vom Mythos zum Logos” is
accepted, after J. Bidez, La biographie d’Empédocle (Ghent, 1894), 150ff, by Kranz, Hermes
70 (1935) 111, Lmped. passim. On the opposite side, for dating the Purifications after the
Ou Nature (favored by fr. 1 31) are Diels (SBBin 1898, 396ff), Wilamowitz (SBBin 1929,
0266), and especially Reinhardt (CP 45 [1950] 170ff). For the inner unity of the two poems
(c.g., fr. 115 has the 4 elements and Strife: fr. 23.11 represents Empedocles as god or nearly
so, fr. 15 has immortality, and in the phrase “‘life, as men call it’’ anallusion to doctrines
of release or salvation), Nestle (Philologus 65 [1906] 545ff), Nestle (in ZN I 1007ff),
Jaeger (Theol. 128ff), Dodds (Irr. 145f, with nn.); H. Schwabl (WS 69 [1956] 49-56),
and esp. Kahn (AGP 1960). Long (AJP 70 [1949] 142-158) tries to make the matter too
neatly systematic.
296
. Fifth-Century Reflections of Pythagorean Philosophy ?
the mark of a very ancient type, that of the shaman! Thus one might
gucss that in Pythagoras, too, both aspects were present—but wisdom
is not science. If we suppose that the fact that the elements are four in
number is related to the symbolic power of the number four? or
that the idea of harmony suggests a relationship to the Pythagoreans,1
still he is not known to be concerned with music theory, or with Limit
and Unlimited; and if he speaks of a ratio of mixture of the elements,
that does not mean that “things are numbers.”” His basic ontological
idea, the characterization of the elements as material that is qualitatively
unalterable, without origin, and imperishable, agrees even in vocabu-
lary with Parmenides; and in astronomy and physiology Empedocles
employs an unabashed empiricism, which takes what it needs from any
source. The doctrine of transmigration comes from Pythagoreanism,
just as some individual cosmological ideas have their roots in myth,}%
but his ontology and natural science are not Pythagorean.
Alcmaeon, Parmenides, Empedocles, Hippasus, Hippo, Philolaus—
all of them sprang from the soil of Magna Graecia at a time when the
doctrine of Pythagoras was still alive; and all of them, as well as the
Ionic book On Sevens, had points of contact with Pythagoreanism.
But to infer in each case the priority of a developed, systematic Pythag-
orean philosophy, is petitio principii. If we consider these persons as a
group, it is almost impossible to reconstruct as their common back-
ground any kind of well-defined scientific and philosophical system.
If we look for a dualistic foundation, which could unite Alcmacon,
Parmenides, and to a certain extent Empedocles, with Philolaus, then
Hippasus and Hippo must be left out. And even aside from this, the
differences are considerable: in Empedocles we have the periodic
interchange of opposite world-states, to compare with Philolaus’
100 Dodds, Irr. 146.
10) Cf. Zeller I gsof; Diels, Elementum (Leipzig, 1899) 15; above, ch. II 4, n. 1 55:.-
Rostagni (Verbo 262f) would like to discover a Pythagorean doctrine of elements; but his
references (including Philolaus fr. 12) are all under the influence of the Timaeus (above,
ch. I 3). To be sure, there is a pre-philosophical conception of “elements especially in
the Indo-Iranian area, in which the cosmos is divided into regions of fire (the stars), air,
water, and earth. (Cf. J. Przyluski, “L’influence iranienne en Gréce et dans Inde,”
Rev. de ?' Univ. de Bruxelles 37 [1932] 283-294; Olerud 136ff). We cannot go into the
question here, to what extent something of the kind existed in Greece (Kranz, Hermes
70 [1935] 113ff, mentions Orphism, but this is a very uncertain factor).
102 Above, n. 93; ch. III 2, n. 65.
108 ‘When Empedocles compares the shape of the world to an egg, we cannot ignore
the relationship of this to Orphic cosmogony (As50, Ar. Av. 693ff, Nestle in ZN I
980 ἢ. 2).
297
σαν δ € 22 GANS bd £4 i
harmony of opposites in a single world. This is comparable with
Alemacon’s ἐσονομία of opposites in the organism; but in Parmenides
the opposites are separate and irreconcilable. If we postulate a philosophy
of form, or of an advanced degree of abstraction, we might set
Philolaus alongside the Elcatics; but the atomists come much closer
to belonging in this company than the sensuous world of Empedocles,
to say nothing of Hippasus and Hippo. There is no single characteristic
trait to unite the astronomical views of Alcmaeon, Parmenides,
Empedocles, and Philolaus; in fact there is one point in which the
latter agrees rather with the Ionians than with Parmenides and Platg.1%
If there is a sense in which Pythagoreanism forms part of the back-
ground for all this, it is not Pythagoreanism as a fully formed scientific
and philosophical structure, but as an inspiration or stimulus of a pre-
philosophical sort, in the area of mythic lore and religious aspiration,
which each, in his own way and in accordance with his own person-
ality, allowed to influence him in the formulation of his thoughts.
This is true not only of Alcmacon, Parmenides, and Empedocles, but
also of Philolaus, whose thoughts of the “limiting,” the “unlimited,”
and of number were helped by the efforts of Plato to a career whose
scope could hardly have been foreseen at the beginning.
There is no law of reciprocal interaction in the field of thought,
which could make it possible for us, by inference, to fill adequately
the gaps in our tradition, as the law of gravitation enables astronomers
to calculate the position or movement of an unknown star. In fact, the
suspicion persists that the lacuna in the tradition about early Pythago-
reanism is not an accident. If we cannot get aclear idea of the philosophy
and science of Pythagoras, it is because Plato and Aristotle did not
consider him a philosopher. If we cannot find a clue to the philosophy
of Limit and Unlimited and their harmony achieved through number,
before the day of Philolaus, it is because this doctrine, in this abstract
form, was first created as Philolaus worked to formulate anew, with
the help of fifth-century φυσιολογία, a view of the world that came to
him, somchow, from Pythagoras.
104 : : : ;
"In the assumption of a material περιέχον outside the heaven (Kahn 234 n. 4)
298
IV, Astronomy and Pythagoreanism
I. THE STRUCTURE OF THE WORLD AND THE PLANETARY SYSTEM
The Greeks acknowledged, almost too eagerly, that Greek astronomy
was based on the accomplishments of the East.1 Modern scholarship,
studying the original Babylonian and Egyptian sources, has made
clear how much of the final achievement is due to the Greeks them-
selves.? It is true that Greek astronomers used the observational data
which by various routes made their way from the Orient, and especially
from Babylonia;? but they made an original contribution, without
precedent in the East, in the development of a conception of the
world’s structure from the insights of Greek mathematics—the famous
Ptolemaic system, in which the planets circle about the spherical carth,
at various distances, enclosed in turn by the sphere of the fixed stars.
The risings and settings of the stars were thought of as related to the
geometry of the sphere, and the irregularities in the paths of the planets
were explained by the combination of mathematically perfect circular
movements. The problem of cosmic distances was taken up too, and
not without success.* From Eudoxus through Hipparchus to Ptolemy,
the development of this Greek scientific enterprise can be followed
fairly easily, but what came before the first great epoch ts, as so often,
difficult to make out clearly.
1 See, for example, (Pl.) Epin. 986e, Arist. Cael. 292a8, Diod. 2.30.
2 See esp. Neugebauer, ExSc 156. It is important that, contrary to a widespread belief,
the Greek planetary system cannot be shown to have been Babylonian; cf. Boll, RE
VII 2561ff.
3 According to Sen. QNat 7.3.2, Eudoxus was the first to bring from Egypt exact
data about the movements of the planets (T15; cf. T12-20 Lasserre). About 500 B.c.
Babylonian astronomy was exerting an influence in Egypt, as is shown by a new discovery:
R. A. Parker, A Vienna Demotic Papyrus on Eclipse- and Lunar Omina (Providence, 1959);
van der Waerden, Anf. 131-133. Also, cf. the reference to ‘‘Egyptian” observations in
Arist. Mete. 343b10, 28; cf. Cael. 292a8; Chaeremon FGrHist 618F7. During Alexander’s
expedition, Callisthenes is supposed to have sent Babylonian data to Greece directly
(Simpl. Cael. 506.11 = FGrHist 12473; cf. Hipparchus ap. Ptol. Syt. 4.11 p. 340
Heiberg). Suspicion is aroused by the late attestation, its novelistic presentation, and the
absurd claim that observations had been carried on for 31,000 years in Babylon (Neu-
gebaucr, ExSc 151).
4 Hipparchus came fairly close to establishing the distance of the moon correctly,
putting it at 33% times the diameter of the carth (RE ΝΠῚ 1676).
299
ΠῚ, PHILOLAUS
harmony of opposites ina single world. This is comparable with
Alcmacon’s ἐσονομία of opposites in the organism; but in Parmenides
the opposites are separate and irreconcilable. If we postulate a philosophy
of form, or of an advanced degree of abstraction, we might set
Philolaus alongside the Eleatics; but the atomists come much closer
to belonging in this company than the sensuous world of Empedocles,
to say nothing of Hippasus and Hippo. There is no single characteristic
trait to unite the astronomical views of Alcmacon, Parmenides,
Empedocles, and Philolaus; in fact there is one point in which the
latter agrees rather with the Ionians than with Parmenides and Platg.1%
If there is a sense in which Pythagoreanism forms part of the back-
ground for all this, it is not Pythagoreanism as a fully formed scientific
and philosophical structure, but as an inspiration or stimulus of a pre-
philosophical sort, in the area of mythic lore and religious aspiration,
which each, in his own way and in accordance with his own person-
ality, allowed to influence him in the formulation of his thoughts.
This is true not only of Alcmaeon, Parmenides, and Empedocles, but
also of Philolaus, whose thoughts of the “limiting,” the “‘ unlimited,”
and of number were helped by the efforts of Plato to a career whose
scope could hardly have been foreseen at the beginning.
There is no law of reciprocal interaction in the field of thought,
which could make it possible for us, by inference, to fill adequately
the gaps in our tradition, as the law of gravitation enables astronomers
to calculate the position or movement of an unknown star. In fact, the
suspicion persists that the lacuna in the tradition about early Pythago-
reanism is not an accident. If we cannot geta clear idea of the philosophy
and science of Pythagoras, it is because Plato and Aristotle did not
consider him a philosopher. If we cannot find a clue to the philosophy
of Limit and Unlimited and their harmony achieved through number,
before the day of Philolaus, it is because this doctrine, in this abstract
form, was first created as Philolaus worked to formulate anew, with
the help of fifth-century φυσιολογία, a view of the world that came to
him, somehow, from Pythagoras.
104 In the assumption of a material περιέχον outside the heaven (Kahn 234 n. 4).
298
ΙΖ. Astronomy and Pythagoreanism
I. THE STRUCTURE OF THE WORLD AND THE PLANETARY SYSTEM
The Greeks acknowledged, almost too eagerly, that Greek astronomy
was based on the accomplishments of the East.1 Modern scholarship,
studying the original Babylonian and Egyptian sources, has made
clear how much of the final achievement is due to the Greeks them-
selves.? It is true that Greek astronomers used the observational data
which by various routes made their way from the Orient, and especially
from Babylonia;? but they made an original contribution, without
precedent in the East, in the development of a conception of the
world’s structure from the insights of Greek mathematics—the famous
Ptolemaic system, in which the planets circle about the spherical earth,
at various distances, enclosed in turn by the sphere of the fixed stars.
The risings and settings of the stars were thought of as related to the
geometry of the sphere, and the irregularities in the paths of the plancts
were explained by the combination of mathematically perfect circular
movements. The problem of cosmic distances was taken up too, and
not without success. From Eudoxus through Hipparchus to Ptolemy,
the development of this Greek scientific enterprise can be followed
fairly easily, but what came before the first great cpoch is, as so often,
difficult to make out clearly.
1 See, for example, (Ρ].) Epin. 986e, Arist. Cael. 29208, Diod. 2.30.
2 See esp. Neugebauer, ExSc 156. It is important that, contrary to a widespread belict,
the Greek planetary system cannot be shown to have been Babylonian; cf. Voll, RE
VII 2561ff.
8 According to Sen. QNat 7.3.2, Eudoxus was the first to bring from Egypt exact
data about the movements of the plancts (T15; cf. T12-20 Lasserre). About s00 B.C.
Babylonian astronomy was exerting an influence in Egypt, as is shown by anew discovery :
R. A. Parker, A Vienna Demotic Papyrus on Eclipse- and Lunar Omina (Providence, 19§9);
van der Waerden, Anf. 131-133. Also, cf. the reference to “Egyptian” observations in
Arist. Mete. 343b10, 28; cf. Cael. 29228; Chaeremon FGrHist 618F7. During Alexander's
expedition, Callisthenes is supposed to have sent Babylonian data to Greece directly
(Simpl. Cael. 506.11 == FGrHist 12473; cf. Hipparchus ap. Ptol. Syut. 4.11 p. 340
Heiberg). Suspicion is aroused by the late attestation, its novelistic presentation, and the
absurd claim that observations had been carried on for 31,000 years in Babylon (Neu-
gebaucr, ExSc 141).
4 Hipparchus came fairly close to establishing the distance of the moon correctly,
putting it at 334 times the diameter of the earth (RE VEIL 1676).
299
HW’. ASTRONOMY AND PY'THAGOREANISM
The earliest connected discussions of astronomical matters are found
in the works of Plato, and it is not merely a coincidence that almost all
the important astronomers of later times were Platonists.® The Greek
idea of the general structure of the world is set forth here in all its
essential features: the earth is spherical and rests, free of support, at the
center of the sphere of the fixed stars: the planets are stationed in
concentric paths at varying distances; and their apparent irregularities
are explained by mathematical principles. The order of the planets,
from the earth at the center, is moon, sun, Venus, Mercury, Mars,
Jupiter, Saturn‘—an order retained by Eudoxus, Callippus, Aristotle,
and even Eratosthenes.’ It is “correct,” insofar as the planets are arranged
according to how long it takes them to make a circuit through the
zodiac, with a longer time corresponding to a greater distance. Since
the inner planets Venus and Mercury are, from the geocentric point
of view, “isodromous” with the sun,® that is, like it they make the
circuit of the zodiac in a year, their position in relation to the sun cannot
be determined on this principle, and this is what led to the ancient
controversies about the order of the planets.
It is known that the facts about the planets came to Greece from
Babylon. They were known centuries earlier there, and it is from there
and not from Greek mythology that we have the association of the
® The Platonism of Eratosthenes and Ptolemy is well known; but Hipparchus, too,
shows himself to be in the influence of the Timaeus when he teaches “‘cognationem cum _
homine siderum animasque nostras partem esse caeli” (Plin. HN 2.95).
ὁ Perspicuously described at Rep. 616e et seq.; Tim. 38d emphasizes the number 7
and names Venus and Mercury, but when the idea is taken up again he reverses the order
and names Mercury first. This is why Aét. 2.1§.4 has Plato putting Venus over Mercury
(followed, no doubt, by Eratosthenes in his Hermes; Chalcid. 73). Epin. 987b has a simple
enumeration; here it is also stated that aside from ἑωσφόρος the planets are nameless,
though ἐπωνυμίαν εἰλήφασιν θεῶν. Here and later the style of appellation is ὁ (τῆς)
᾿Αφροδίτης ἀστήρ... .; the names Φωσφόρος, Στίλβων, Πυρόεις, Φαέθων, Φαίνων ἀο ποὶ
occur before the Hellenistic period (Heraclides fr. 66 is of doubtful value; cf. Wehrli 83).
Not before the first century B.c. is a planet called simply Aphrodite or the like (Cumont,
AC 1935, 5-43). On Plato’s astronomy see the commentaries on the Timaeus; Duhem
28ff; Heath, Aristarchus 1 34-189; Math. 310ff.
* Eudoxus: Procl. In Tim. ΠῚ 62 (cf. Eudox. Ars astr. 22) = Do Lasserre; Kenocrates
fr. 17. Callippus—Aristotle: Met. 1073b17ff, Procl. loc. cit., De mundo 392a23ff. Chrysippus:
SVF Π no. 527. Eratosthenes Hermes: Adrastus ap. Theo Sm. 142.7 == Chalcid. 73;
“Pythagoras” Anon. Phot. 439b19ff, Cic. Nat.d. 2.$2f (Stoic praise of cosmic order);
IG XII 1.913 (100 B.c.); Pap. Aberdeen 13 (Aug. I, A.D. 187). This order of the planets is
also to be assumed for Heraclides Ponticus fr. 054 (so that the sun is not central, as Wehrli
92f supposes; the spheres assigned to Pluto are, aside from that of the moon, those of the
elements; cf. Anon. Phot. 430b25 and Hebd. 1).-—On the whole subject, Immisch 6off,
and esp. Boll, RE VIL 25664f.
* ἰσόδρομοι, Tim. 38d, Macrob. Som. Sc. 1.19.4, ete.
1, The World and the Planetary System
planets with individual gods, which still provides their names.® The
discovery of the most important data on planetary movement is due
to the Babylonians, and in particular the time of their orbits. At least
in the case of Saturn, whose orbit takes 294 years, this would require
several generations. The discovery of the planet Mercury, which is
difficult to observe, also belongs here; in fact, a fund of exact knowledge
about the planets is doubtless the most obvious debt to oriental knowl-
edge in the pre-Platonic period. |
Along with knowledge of the planets, the recognition of the spheri-
cal shape of the earth and the postulate of perfect circular movements
make up the world picture of Greek astronomy, as it is presented by
Plato. Ever since the ancient commentators on the Timaeus it has been
thought that “the’’ Pythagoreans were the source to which Plato owed
his astronomical interest and knowledge, and this is not the least
important cause of the high place accorded them in the history of
science.
Aristotle, however, records, along with the famous ideas of the
harmony of the spheres and the special theories of “some” Pythag-
oreans about comets and the Milky Way, another peculiar Pythag-
orean system. Here the earth is one of the planets, and circles, along
with a “counter-earth’’ which is invisible to us, about a “central
fire.” The doxographers attributed this system to Philolaus, and we
may justifiably use the well-established expression, “the system of
Philolaus,” without prejudice to the question of its date or its relation
to the Pythagorean named in the Phaedo. For in fact this relation has
come into question among modern scholars. The attribution to
Philolaus of the system described by Aristotle suggested the recon-
struction of an older, more simple, geocentric system, supposedly
developed by Pythagoras himself or the early Pythagoreans; the
materials for this enterprise were some late references, but, above all,
®See Boll, RE VII 2561ff; Gundel, RE XX 2025, 2029; Meissner 405ff; van der
Waerden, Anf. 105, 108, 172. The Babylonian gods are Ishtar, Nabu, Nergal, Marduk,
and Ninurta (Aphrodite, Hermes, Ares, Zeus, Cronus). The designation of Saturn as
ἡλίου ἀστήρ is Babylonian (Pl. Epin. 987c; the text is emended even by Burnet; sec,
against this, Eudox. Ars astr. col. V, Diod. 2.30, Simpl. Cael. 495.28, J. Bidez, RPh 29
[1905] 319f, Mcissner 254). The colors attributed to the planets in Pl. Rep. 616e go back
to Babylonia (Bidez, Fos app. 1), in spite of the objections of W. J. W. Koster (Le mythe
de Platon, de Zarathoustra et les Chaldéens |Leiden, τὸς 1] 66ff), who thinks this is based on
independent observation. For example, Jupiter, called λευκότατος by Plato, is simply
called ‘the white star’? (Molobabar) in Babylon (Meissner 404).
ἐγ, ASTRONOMY AND PYTHAGOREANISM
conjectures based on Parmenides and Plato.’ Then Frank declared the
Philolaic system too bold and advanced even for the end of the fifth
century and dated it in the years of Plato’s old age? For him, the
history of astronomy was one of the most striking proofs of the
spuriousness of the Philolaus fragments. These arguments, however,
must be reexamined, From the point of view of the history of science,
the most important points are the discovery of the spherical shape of
the earth, the recognition of the five planets, and the explanation of the
apparent irregularities of their courses by means of circular movements.
The history of the exact sciences is of special importance in the
history of thought, because, as O. Neugebauer has put it, “τῆς inherent
accuracy of the mathematical sciences will penetrate to some extent
into purely historical problems.’’2 For this reason it is a special
methodological advantage that the investigation upon which we are
now embarking can be conducted independently, in all important
respects, of the conclusions reached in earlier chapters.13 But it is very
important to establish at the outset the extent, and thus also the limits,
of the mathematical exactitude to be attained in the history of science.
With great confidence, Frank reconstructs the main stages of Greek
astronomy, in an apparently necessary order:! development of the
understanding of space, of solid geometry and perspective, by Anaxag~
oras and Democritus; discovery of the sphericity of the earth and the
“truce movements of the planets” in the form of “geometrically
perfect orbits” (28) by the Pythagoreans of Archytas’ circle; first
mathematical explanation of the movement of the planets, by Eudoxus;
discovery of the rotation of the earth on its axis, and finally the
“Copernican view of the world” in the system of Philolaus (35)—
which thus represents the second step beyond Eudoxus. Similarly, van
der Waerden'® sces in “the logical development of astronomy” the
' Gruppe, KesmSyst sof; Boeckh, KosmSyst 89; and esp. Martin, Pyth. Do6ring already
was calling the theory “generally recognized” in AGP 1892, 508; cf., i.a., Burnet EGP
bof, 296; Berger, Erdkunde τς, Heath, Aristarchus 48ff; van der Waerden, Astr.
20ff; Guthric I 280ff.
"Below, ch. IV 3.
"“ νδε ας
"What we shall presuppose, from previous discussion, is a fundamental skepticism
of the post-Aristotelian tradition, strongly influenced by Platonism, as well as the relative
priority of the actismata—and both of these points are confirmed again and again. We
may adda few points of detail, like the dependence of the so-called “more genuine”
Pythagoreans on Aristotle (ch. III 1) and the (related) analysis of Philolaus Α τό (ch. ΠΙ 2).
™ Brank off; cf. 184ff.
' Astr, 28. Yet, in agreement with the ancient sources, he has the theory of the revolu-
tion ofthe earth on its axis originating in the Philolaic system, while Prank has it preceding
the more complicated one.
302
1. The Worid and the Planetary System
“strongest proof” that the system of Philolaus was preceded by that
geocentric, Pythagorean system which must be assumed, Still and all,
in the history of science logical necessity and historical sequence are
not always identical. Of course, every forward step in this field depends
on a certain group of preliminary studies, but it is just as obvious that
there do occur backward steps, significant enough to cause correct
answers already found to be given up.!® And intuitive anticipations
occur, too, based on inadequate foundation, so that, in such a case,
true progress consists in giving up results which, to hindsight, will be
seen as correct. Perhaps the allegedly “Copernican” system of Philo-
laus will fit into this category.” |
Here, as always, that evidence must be decisive which gives us a
clue to the purely fortuitous aspects of the development; for if we were
to rely on logical calculation of the probabilities, we should find a
good many alternative possibilities. Above all, an inference about
preconditions, though it have almost the certainty of mathematical
demonstration, can never, at the same time, prove who is responsible
for these earlier achievements. That the astronomy which preceded
the Philolaus system was Pythagorean, or that Pythagoras set up an
astronomical system of the same or even a higher order than those
devised by Anaximander or Parmenides, is nothing but a historical
hypothesis, which cannot be corroborated or refuted by the uae
logic of the history of science, but must be known from externa
testimony. For the most part, this question is not even considered,
under the influence of the Platonic tradition all mathematical science
in early Greece is called Pythagorean2* To put the question at all
means to remove one of the main supports from under most of the
reconstructions.
Late sources attribute to Pythagoras a decisive role in the formation
of the Greek view of the world’s structure, but these reports are
strikingly contradicted by statements about Parmenides and the latter
has the oldest authorities on his side. The matter in question is the
16 As compared with Anaximander, the astronomical views of Xenophanes a
Heraclitus are regressive (Heath, Aristarchus 54ff); but so are those of Anaximenes, ᾿ ο
does not dare to let the earth hang ge Ἢ space. ie os pores of Aristarchus,
> i ee below, ch. IV 2, n. 1; ch. ,n.t.
εν τι eae eee ‘to mention the Philolaic system, modern scholars
would confidently date it in the era of Aristarchus of Samos! . ;
18 Even Frank clings to the Pythagorean origin of Greek natural science and mathe-
matics, only he dates it in the time of Archytas.
303
IV. ASTRONOMY AND PY'THAGOREANISM
sphericity of the earth and the identity of evening star with morning
star, which means the beginning of a true understanding of the planets.
Diogenes Lacrtius writes of Pythagoras: τὸν οὐρανὸν πρῶτον ὀνομάσαι
κόσμον καὶ τὴν γῆν στρογγύλην' ὡς δὲ Θεόφραστος, Happevidny?® In
another passage, which in all likelihood goes back to Theophrastus,
he likewise says of Parmenides, πρῶτος δὲ οὗτος τὴν γῆν ἀπέφαινε
σφαιροειδῆ καὶ ἐν μέσῳ κεῖσθαι.20 The spherical earth appears again
as a doctrine of Pythagoras in the Hypomnemata (reported by Diogenes
Laertius, 8.25).
At first, the natural assumption seemed to be that Parmenides had
learned this fact from Pythagoras and published it,24 but Frank main-
tained vigorously that Parmenides could not have known that the
earth was spherical; it could be seen from Plato’s Phaedo, he thought,
that at the time of that dialogue this was a quite new discovery.
Parmenides had, according to Theophrastus’ evidence, merely been
the first to use the word στρογγύλος for a disc-shaped carth.22 And
recently it has even been contested whether Plato does describe the
carth as a sphere in the Phaedo.28
But the Theophrastus passage cannot be eliminated so easily.24 We
cannot, in fact, ascertain what word Theophrastus used, but it seems
sure that in the context he was thinking of a spherical rather than a
discoid body,?* so that the dubious inference from the Phaedo cannot
be used against the evidence of Theophrastus.”
1915. L. 8.48 = Theophr. Phys. op. fr. 17, Dox. 492 = DK 28Aqq4.
” [D. L. 9.21 = Theophr. Phys. op. fr. 6a, Dox. 482; on the attribution to Theophrastus,
Dox. 166ff.
41 Tannery, HScH 236.
Ὧ5 Frank 184ff, 198ff; Heidel, Maps 7off, 816, Morrison, JHS 1955, 64.
ἍΤ G. Rosenmeyer, CQ 50 (1956) 193-197; W. M. Calder, Phronesis 3 (1958)
121-125; Rosenmeyer, ibid. 4 (1959) 71-72; Morrison, ibid. 101-1 19. The myth may
preserve a bit of older cosmology, but Phd. 1126 seems to refer to the middle of a sphere.
Theophrastus’ testimony on Parmenides helps to decide the issue.
" Against Frank, P. Friedlaender, Plato I (New York, 1958) 386; Mondolfo, “La
prima affermazione della sfericita della terra,” Accad. d. scienze di Bologna 1937 (not avail-
able to me; see ΖΜ 339ff); Rehm-Vogel 12; Kahn 115ff; cf. Thomson r1off.
"The word orpoyyvaos is used with reference to Pythagoras, and here surely means
“spherical.” The parallel passage D.L. 9.21 has σφαιροειδής, which is of course not reliable.
(At ID. L. 2.1, the earth of Anaximander, the “column drum,” is called σφαιροειδής,
as Frank remarks, 200.) L'rpoyyvAos can mean circular or spherical; it is used of hailstones
at Aristoph. Nub. 1127. It is not certain what shape is attributed to the earth by Diogenes
of Apollonia; it is called στρογγύλη in D.L. 9.57.
5 At Pl. Phd. 108e, the word περιφερής (στρογγύλος, 976) is used quite casually; the
main thing is the assurance that the earth docs not need the support of air or the like:
Plato is here using another doctrine to hark back, against Anaxagoras (A88), Diogenes of
Apollonia (A16a; cf. C2 = Eur. Tro. 884), and Democritus (Arist. Cael, 294b14f; also
Anaximencs A6, 20), to Anaximander and Parmenides.
304
ι. The World and the Planetary System
Not only was the Democritean Bion of Abdera acquainted, about
400, with the mathematical consequences of the spherical shape of the
earth,?”? but about 430 8.c. Hippocrates of Chios had projected the
celestial circles onto the carth, obviously presupposing its sphericity.?* A
spherical earth is but an easy step from the thesis that it is located in
the middle of the universe and “because of equality” did not fall in
one direction or the other. This is a thesis Parmenides took over from
Anaximander; and if the description of the world of doxa is an indepen-
dent creation of Parmenides, it is not at all far fetched to suppose that,
having in mind the perfect “sphere’’ of Being, he was the first to call
the earth on which we live a sphere.?® Empedocles seems to have
followed him.*° That this theory did not win out, and that Democritus
clung to his peculiar idea of a bowl-shaped earth,*! is one of those
retrograde steps that sometimes occur in the history of the natural
sciences.
Posidonius found that Parmenides was the ἀρχηγός of the division
of the earth into five zones, but objected that he had made the torrid
zone too broad, so that it extended beyond the tropics.5* Most scholars
have been very skeptical of this datum,** but it fits quite well into the
Parmenidean dichotomy of Fire and Night: just as, in the heavens,
Fire and Night occur both pure and mixed, so on the surface of the
27 DK 77.1 = DL. 4.58: οὗτος πρῶτος εἶπεν εἶναί twas οἰκήσεις, ἔνθα γίνεσθαι ἕξ
μηνῶν τὴν νύκτα καὶ ἕξ τὴν ἡμέραν. Strabo calls him ἀστρολόγος (1, p. 29 = DK 77.2).
Wellmann (RE III 486) dates Bion, rightly, before Eudoxus. Bion wrote Ionic and Attic
(DK 77.1). We need not decide whether older myths may have contained reminiscences
of the long polar nights (cf. the controversies over Od. 10.82ff and Hd¢t. 4.25).
28 DK 42.5 = Arist. Mete. 343a8: the torrid zone between the tropics. On Hippocrates’
date, see below, n. 77.
29 The earth at the center: A44, following Anaximander (below, ἢ. 44); Being com-
pared to a sphere: fr. 8.43; cf. Theophr. Phys. op. fr. 6, Dox. 482. See Gigon, Ursprung
86f, 275.
80 Kranz, Emped. 50: “die Erde nach damaliger, altpythagoreischer Vorstellung als
Kugel im Mittelpunkt” (but cf. below, n. 44). Actually, the idea (A56) that the sun is a
reflection of the “round” earth (κυκλοτερής) implies sphericity, since the sun is not
stationed directly over our section of the earth, and the reflection of a disc would appear
distorted rather than perfectly round. But it is naturally an open question to what extent
Empedocles was preoccupied with such geometrical considerations. He did give up the
idea of Hades as a region under the earth (below, ch. IV 4), and this would be consistent
with a spherical earth.
$1 Aga; Frank 187ff. Oriental tradition may be behind this. According to Babylonian
teaching the earth is σκαφοειδὴς Kai κοίλη (Diod. 2.31; Meissner 107ff). Herodotus ignores
the sphericity of the earth (e.g., 3.104). On Hebd. 2, above, ch. III 3, n. 63.
82 Posidonius ap. Strabo 2, p. 94 = DK 28A44a; derived from this, Ach. Is. p. 67.27
Maass, Aét. 3.11.4 (where, in contradiction of Posidonius, a connection with the ζῶναι
τροπικαί is claimed—a simplification).
88 Reinhardt, Parm. 147 n. 1, Kosmos 361 ἢ. 2; Heidel, Maps 76, 80, 91; cf. Rehm-
Vogel 11f,
395
ἐγ, ASTRONOMY AND PYTHAGOREANISM
earth there is amixed zone between the extremes; and, as the ‘‘circles”’
in the heavens were arranged in some kind of symmictry, the southern
hemisphere is a symmetrical reverse of the northern. And the very fact
that, according τὸ Posidonius, no mathematical astronomy was
associated with this, or any projection of the tropics onto the earth, is
an encouraging sign.§4 On the other hand, the doxographers ascribe
to Pythagoras the precise astronomical division of the earth into five
zones, bounded by the tropics and the polar circles,3® ἀναλόγως τῇ τοῦ
παντὸς σφαίρᾳ, and trace the corresponding division of the celestial
sphere to Θαλῆς Πυθαγόρας καὶ of ἀπ’ αὐτοῦ͵ 36 and the determination
of the obliquity of the ecliptic, upon which the location of the tropics
of course depends, to Pythagoras himself, adding ἥντινα Οἰνοπίδης ὃ
Χῖος ὡς ἰδίαν ἐπίνοιαν σφετερίζεται 37 We have a parallel for this last
sentence in the words of Aristotle’s pupil Eudemus, the best authority
for the history of Greek science before the Hellenistic period: Οἰνοπίδης
εὗρε πρῶτος τὴν τοῦ ζῳδιακοῦ Sidtwow ... While Eudemus appears
have spoken simply of a discovery of Oenopides,*® the later tradition
maintains that this is a plagiarism from Pythagoras. For both the
sphericity of the earth and the division of it into zones, Pythagoras’
name stands in competition with Parmenides’; in the second case the
more advanced, “correct” formulation of the theory is ascribed to
Pythagoras; and in both cases the earlier evidence—that of Theo-
phrastus and Eudemus—has no mention of Pythagoras.
54 On the problems of Parmenides’ celestial system, see below, n. 40. Frank (200 n. 1)
wrongly states that Posidonius “attributed to Parmenides the projection of the tropics
from the celestial sphere onto the concentric spherical earth.” Posidonius’ words were
(Παρμενίδην) σχεδόν τι διπλασίαν ἀποφαίνειν τὸ πλάτος τὴν διακεκαυμένην... .]
ὑπερπίπτουσαν ἑκατέρων τῶν τροπικῶν εἰς τὸ ἐκτὸς... According to this the width of
the torrid zone would be not 46° 54’ 6” but about 90°, i.e., for Parmenides it is a “ring”
whose width is about a fourth of its circumference; and this has nothing to do with
spe or oe (differently Hippocrates, above, n. 28).
Act. 3.14.1; cf. Mart. Cap. 6.600 (MS
tiene ἢ p 9 (MSS pytharas or pythagoras, wrongly corrected,
= Aét. 2.12.1; cf. 2.23.6: Πλάτων Πυθαγόρας ᾿Αριστοτέλης (on the obliquity of the
ecliptic), where the position of the name Pythagoras is a giveaway. |
47 Aét. 2.12.2.
7 Eudemus fr. 145 = Theo Sm. 198.14f. Diels (DK 41.7) conjectured λόξωσιν for
διάξωσιν. The topic of discussion is the measurement of the angle of the ecliptic; the fact
of its obliquity—i.e. of the zodiac—was known since Anaximander (A5) and Cleo-
stratus (LOK 6B2); see von Fritz, RE XVII 2260f. Of course, Theo gives only a much
abbreviated excerpt from Eudemus; but, if his original had corresponded to the report
of Actius, the name Oenipides would have fallen out rather than Pythagoras.— Aét
2.32.2 ascribes the s9-year cycle to ‘“Oenopides and Pythagoras,” while Ael. VH 10.7
and Cens. 19.2 speak only of Oenopides. The mention of Pythagoras is to be explained
either from the reproach of plagiarism or as a conclusion from Philolaus A22.
306
ι. The World and the Planetary System
Favorinus records the same rivalry for the discovery that the evening
and morning stars are the same.3® This planct—did he already call it
after Aphrodite?—had a special meaning for Parmenides, and he has
it circling in the pure αἰθήρ above the sun. This double-track tradition
about Parmenides and Pythagoras can be understood in the light of a
statement of Diogenes Laertius about Parmenides: “He seems to have
been the first to discover that the evening star and the morning star are
the same, as Favorinus says ... but some attribute this to Pythagoras;
but Callimachus denies that he is the author of the poem.”’ Callimachus
denied Pythagoras’ authorship of a certain poem which dealt, among
other things, with the planet Venus.*! Actually, Heraclides Lembus, in
his list of the writings of Pythagoras, gives first place to a work epi
τοῦ ὅλου ἐν ἔπεσιν (D.L. 8.7). It is likely that this poem dealt not only
with the heavenly bodies but with the spherical shape of the earth and
its division into zones, borrowing from Parmenides, and certainly also
from Empedocles. No one will claim that there was a didactic poem
actually written by Pythagoras. Callimachus was right in rejecting the
poem, and in doing so joins Theophrastus and Eudemus as a third
witness against the “Pythagoras” version of this tradition. We need
not be concerned whether the doxography took its reports from the
apocryphal poem or perhaps from older statements by Platonists, for
39D. L. 9.23 (below, n. 41); Parmenides also Aét. 2.15.7; Pythagoras, Apollodorus
FGrHist 244F91 and Plin. HN 2.37 (dated 612 B.c. as a discovery of Pythagoras). It is
almost incredible that so elementary an astronomical fact remained for so long unknown
to the Greeks; about 1580 B.c. the Babylonians already knew the time of an orbit of
Venus (van der Waerden Anf. 49). Wilamowitz, Hermes 18 (1883) 416-423, found the
identity of morning and evening star referred to in mythology; and Ibycus of Rhegium
(approximately a contemporary of Pythagoras; Burnet, EGP 191 n. 3) spoke of it (fr.
331 Page).
40 A4oa. In general, it is regarded as an almost hopeless task to reconstruct the celestial
system of Parmenides (fr. 12; A37). See Raven, KR 284f; De Vogel, GP I 41ff; among
older writers, Tannery, HScH 238ff; Zeller I 7o8ff; Burnet, EGP 187ff; Gigon, Ursprung
276ff; a quite different answer by Morrison, JHS 1955, 6off. Reinhardt (Parm. 11ff)
is probably right that it was primarily a cosmogonical process (which would be under-
standable in connection with Anaximander Alo and Empedocles A30, 49, 50); but the
result must be our world.
417). L. 9.23, Callim. fr. 442 Pfeiffer; cf. Wilamowitz, Platon II 85.2. Pfeiffer thinks
of a poem whose authorship was contested between partisans of the two men; but there
was only one book by Parmenides (D.L. 1.16 = DK 28A13). Diels (DK I 225, n.) thought
that this one didactic poem of Parmenides was circulated under Pythagoras’ name, but
it is more likely that there was an apocryphal revision labeled “ Pythagoras.”—D. L.
8.14: πρῶτόν τε ἕσπερον καὶ Φωσφόρον τὸν αὐτὸν εἰπεῖν (Πυθαγόραν) ws φησι Παρμενίδης.
This has been emended from the time of Casaubon to οἱ δέ φασι Παρμενίδην. Diels,
IDK 28A40a considers ὥς φησι (καὶ) Παρμενίδης. Burnet, EGP 191 n. 3, defends the MS
text. But, if Parmenides had mentioned Pythagoras or even made a clear allusion to him,
the ancient scholars, in their search for evidence about Pythagoras would have preserved
the verse as they did Empedocles fr. 129.
307
Ve ASTRONOMY AND PYTHAGOREANISM
example in the commentaries on the ‘Timacus;*? in neither case are the
statements reliable,
For there is no corroboration in the astronomical field, more than
anywhere else, that Parmenides got his philosophy of nature from a
Pythagorean source. The later tradition ascribes to Pythagoras a much
more advanced system than Parmenides had.? In_ his astronomy,
Parmenides holds quite closely to Anaximander, and Empedocles
closely follows him ;“ there is no reason to inserta Pythagorean science.
Commenting on a passage in which Aristotle refers to the profes-
sional astronomers for the question of the order and the distances of
the stars, Simplicius says, “ Anaximander was the first to discover the
inquiry into the sizes and distances (of the stars), as Eudemus reports,
adding that the Pythagoreans were the first to give their order,’’4
Here is ascribed to the Pythagoreans—though not to Pythagoras*®—a
not unimportant contribution to the development of the world model
found in Plato. Eudemus is trying to discover, in good Aristotelian
fashion, the nature of the progress in science that led to the situation as
he knew it; results are what he wants to record.4? In pursuit of this
aim, he attributed to the Pythagoreans the order of the planets known
to Plato, Aristotle, and Eudoxus. But in order to estimate the period to
which this takes us back, and to understand the relation to the datum
about Anaximander,4® we must survey the theories of the pre-Socratics
on the order of the planets.49
42 The attribution to Pythagoras of the invention of the word κόσμος, mentioned along
with the sphericity of the earth at D.L. 8.48, may well come from Heraclides (above
ch. I 3, ἢ. 1§1).
48 Above, nn. 34-35; cf. ch. IV 2.
44 Above, nn. 29, 40, $8. For the rings of fire and the στεφάναι, ch. III 3, ἢ. 25. Em-
pedocles, too, speaks of “wheels,” at least in a simile (fr. 46).
“Simpl. Cael. 471.1 (on Arist. Cael, 291a29) = Eudemus fr. 146 = DKi2Arg:
... Avagiudvdpov πρώτου τὸν περὶ μεγεθῶν καὶ ἀποστημάτων λόγον εὑρηκότος, ὡς
ϑύδημος ἱστορεῖ τὴν τῆς θέσεως τάξιν εἰς τοὺς ΠΠυθαγορείους πρώτους ἀναφέρων.
oes ΓΙ f ΕΣ “, » ᾿
(The key words τάξις and ἀποστήματα are from Aristotle; θέσεως τάξις is also found in |
Theo Sin, 181.1).
. “ Eudemus, like Aristotle, does not mention Pythagoras in a scientific context. (On
fr. 133, see below, ch. VI 1, n. 62.)
“Cf. πρῶτος frr. 144, 145, 147, 148; and twice in the fragment cited; similarly,
Theophrastus, above, nn. 19-20.
“ The juxtaposition of the two reports is at first hard to understand, since the calcula-
tion of distances and sizes naturally presupposes a certain definite order. This is why
Gundel (RE XX 2041) rejects the statement about Anaximander, and Wehrli (121)
understands Eudemus to be putting the Pythagoreans chronologically ahead of Anaxi-
mander. The source of misunderstanding is often an imprecise translation. E.g., Tannery
(Astr. 126) has “. .. les premicrs ἃ spéculer sur la question de l’ordre des sphéres,” and van
der Waerden (Astr. 29) “. . . die Untersuchung der Reihenfolge ihrer Lage fiihrt er...
auf die Pythagorcer . . . zuriick.’”” Eudemus has nothing about “speculations” or “in-
vestigation”; he is talking about a “finding.”
“ See the material collected by Gundel, RE XX 2038-2046.
308
ι. The World and the Planetary System
The fundamental points of that specifically Greck conception of the
world’s structure are already to be found in Anaximander:®° the earth
remains in place, without support, held by the “equality” of distance
in all directions,®! in the center of the circling rings of fire which are
the paths of the stars.52 Mythical conceptions of the “roots of earth,”
and of Helios floating back over Ocean to his starting place, have been
abandoned ;*8 and we find the idea of a perfect circle playing its role in
astronomy before Pythagoras.*4 Into this bold outline Anaximander
inserts precise figures: the sun is as large as the earth; its “‘circle’’ is
27—or 28—times as large as the earth; that of the moon is 18—or 19—
times as large; and doubtless the corresponding numbers for the stars
would be 9 and 10.55 The remarkable prominence of the number 9 is a
point of contact with Hesiod,®* and the idea that the stars are closest
to the earth, while the sun is furthest away, comes from Iranian
teachings about the soul.*’ In this point Anaximenes introduced the
essential correction, based on consideration of physical phenomena;
50 Tannery, HScH 9goff; Zeller I 297ff; Burnet, EGP 62ff; Diels, AGP 1897, 228ff:
Heath, Aristarchus 31ff; Gigon, Ursprung 84ff; Raven, KR 131ff; Kahn 75ff.
51 A26, A1lI §3. Heidel, Maps 68f, 151, doubts Aristotle’s statement, on the ground
that Anaximander’s cosmos was not a sphere (below, n. 54). But the argument applies
also to circles, rings, or wheels. Aristotle (Cael. 29sbi11ff) is dealing expressly with the
αἰτίαι for the μένειν of the earth. Anaximander must have stated a reason, so that this
cannot be simply a false deduction of Aristotle.
52 Arr §3; Ar8, Aar, Α22.
53 “Roots of the earth,” Hes. Op. 19, Xenophanes fr. 28; the return of Helios, Mim-
nermus fr. 10, Stesichorus fr. 185 P.; Anaximenes A7 §6, Arq returns to the older
idea. Xenophanes (A33, A41) and Heraclitus (fr. 6) are not interested in mathematical
astronomy.
54 It is hard to decide whether a solid shell of the universe is presupposed—a perfect
sphere (as Gigon believes, Ursprung 85)—or whether we look out between the “ wheels”
into the ἄπειρον (as Burnet, EGP 69). The solid shell is part of the mythic background
(χάλκεος οὐρανός) and is also found in Anaximenes (κρυσταλλοειδές, A14), Empedocles
(A30, $1), Hebd. 6 (ἄκριτος πάγος), Hippoc. Vict. 1.10 (6 περιέχων πάγος). Cf. also
Etym. magn. s.v. βηλός.
ὅδ Arr ὃς (incomplete), Α2ι, A22. The numbers for the stars have been reconstructed
since Tannery (HScH 94f). The two sets of figures are generally interpreted as an indica-
tion of the “thickness” of the rings—which must correspond to the size of the earth,
if the sun is the same size as the earth, so that one time the outer, and another time the
inner, diameter would be intended, though this involves an error in computation (Raven,
KR 136 ἢ. 1).
88 Theog. 722f (cf. the Odyssey’s ἐννῆμαρ φερόμην). See Nestle, ZN 301 n.; Kranz,
Kosmos 13f, etc. (9 is an augmentation of 3, which is itself a symbol of plurality; below,
ch. VI 4). Diels (AGP 1897, 230ff) mentions similar ideas among the shamans. Kahn
(94-97) is hardly right in denying these mythical elements in Anaximander.
o7 R. Eisler, Weltenmantel und Himmelszelt 1 (Munich, 1910) 90 n. 3; cf. Boll, RE
VIL 2565; Kranz, NGG 1938, 156; Cumont, Lux 143; Burkert, RhM 1963, 97-134.
Diod. 2.30.6 says that the doctrine is Babylonian; probably it is a matter of Babylonian-
Iranian syncretism. (The Babylonians had a different doctrine of 3 heavens over one
another: Meissner 108.) Kahn (yo) follows Dicls (AGP 1897, 229f) in believing that
Anaximander thought the matter out independently, concluding that the largest fire is
the highest.
100
IV. ASTRONOMY AND PY'THAGOREANISM
the stars, as distinguished from the sun, do not provide us heat, and
are therefore further away (διὰ τὸ μῆκος τῆς ἀποστάσεως) .58
This makes Eudemus’ report on Anaximander comprchensible.
The arrangement of the celestial bodies, ἡ τῆς θέσεως τάξις, is wrong, in
his account, but still he “discovered” a fundamentally important fact.
Not only did he consider the question of the sizes and distances of the
heavenly bodies, but, in spite of the arbitrariness of his hypotheses
and the incorrcctness of his results, he did find the right path toward
the answer. The sun is the same size as the earth, although it looks to
us about a foot wide,5® because it is tremendously far off and apparent
size decreases as distance increases. True size, apparent size, and distance
stand in a definite mathematical relationship. Thus the basic ideas of
geometrical proportion and optics have been “discovered,” and are
then boldly applied to cosmic magnitudes and distances, where any
kind of verification is impossible. This is the λόγος which, according
to Eudemus, Anaximander “was the first to discover.”
What the Pythagoreans could contribute in addition to this was,
after the correction made by Anaximenes, the correct order of the
planets. The planets do not seem to have been discussed by Anaxi-
mander.°° Of course, morning and evening star had been known for a
long time, and people must have noticed long before this that some
other bright stars are not always associated with specific constellations,
but “wander about.” Thus the word itself, πλάνητες (ἀστέρες) may be
old.®! But the further details—that there are five planets, that they have
definite orbital periods and thus also regular courses, and that in the
character of their movements they are to be compared rather with the
sun and moon than with the fixed stars®’—all this came to the Greeks
from Babylon, and later than Anaximander, though before Plato.
Detailed knowledge of the planets cannot be proven for any of the
δ Anaximenes A7 §6. The sun is found again as the highest star in Parmenides (A4oa;
doubted by Tannery HScH 240, Zeller 714 n. 2; but the directly quoted, short fragments
10-12 do not provide any firm guidance on the question; cf. Heath, Aristarchus 74f),
Empedocles (Aso; cf. above, nn. 29, 40, 44 on the line Anaximander—Parmenides-—
Empedocles), Leucippus (Ar 833), Metrodorus (DK 7O0AQ), Crates of Mallos (Aét.
2.1§.6), and in mysteries of late antiquity (Julian Or. 4.148a).
Cf, Arist. De an. 428b2, Heraclitus fr. 3.
"Cf Gundel, RE XX 2041. The fact that Aét. 2.15.6 also speaks of planets (Anaxi-
tmander Ars) proves nothing, since Metrodorus and Crates of Mallos are also included
in the Jemma, Diels supplies the word “planets” as part of a conjectural restoration in
Att (Hippol. Ref. 1.6.5); but this remains uncertain.
*! On its formation sce Schwyzer I 499. The word is applied not only to wandering
stars, but to fevers (πλάνητες πυρετοῦ).
"The natural and unsophisticated way of grouping the heavenly bodies has been—
to the present day-—sun, moon, and stars (Anaximander, Parmenides fr. 10, 11, Hebd.
1.2, 6.1, Anaxagoras fr. 12). In this context one speaks of five planets (of πέντε ἀστέρες,
110
ι. The World and the Planetary System
older pre-Socraties. ‘lo be sure, in Anaximenes the basic distinction
was drawn between the planets and the fixed stars. The stars are
fastened “like nails” in the “ice-like” vault of the sky, though some
float on the air like big leaves.68 Alcmaeon’s view may have been
similar to this,64 and Empedocles’ certainly was.®° Anaxagoras sup-
posed there was an indefinite multitude of planets and explained
comets as produced by the collision of two of them.®* The group of
five planets is still not canonical, and the planets are still not accepted
as equal members in the cosmic hierarchy.
Indication of an important further step is found in Philolaus, but
Arist. Mete. 343431; cf. Aratus 454). It was a scientific achievement not to be underrated
to go against appearances, separate the five planets from the fixed stars, and classify them
with the sun and moon, so that there are seven planets: ἑπτὰ κύκλοι, Pl. Tim. 36d, 38c-d;
ot ἑπτὰ ἀστέρες, Schol. Arat. p. 429.12, 478.8 Maass; οἱ ἀστέρες of ἑπτὰ, Dio Cassius
37.18.1; τὰ ἑπτὰ ἄστρα, Hippol. Ref. 1.2.2; ‘‘septem sidera,” Manil. 1.308; Plin. HN
2.12; cf. Hymn. Hom. Ares 7, Cic. Rep. 6.17, etc. Of course this is correct, from the geo-
centric point of view.
88 A7 = Hippol. Ref. 1.7.4: ἥλιον καὶ σελήνην καὶ τὰ ἄλλα ἄστρα πάντα πύρινα ὄντα
ἐποχεῖσθαι τῷ ἀέρι διὰ πλάτος. Cf. Ars, but A14 = Aét. 2.14.3: ἥλων δίκην καταπεπηγέναι
τὰ ἄστρα τῷ κρυσταλλοειδεῖ. The contradiction can be removed by understanding the
first of the sentences quoted of the planets, and the second of the fixed stars. A slight
alteration in the text of the next clause in Aétius (2.14.4) would yield this sense: ἔνια
(MSS ἔνιοι) δὲ πέταλα εἶναι πύρινα ὥσπερ ζωγραφήματα (Heath, Aristarchus 42). Ach.
Is. p. 40.20 Maass, however, has τινὲς δὲ in the corresponding clause; the source of the
trouble must be older. Cf. Gundel, RE XX 2042; Guthrie I 135-137.
64 Ag = Aét. 2.16.3 may originally have applied only to the sun, since Alcmaeon’s
name appears only as an afterthought. In A1z ἀστέρες are named along with the οὐρανός,
but it is not really necessary to interpret this as indicating a distinction between planets
and fixed stars. In any case, Alcmaeon would not have been the first to speak of the planets
(as Gigon says, Ursprung 150), but following Anaximenes, with whom the doctrine of the
“flat” sun also brings him into contact (A4; cf. Anaximenes Α15).
85 As4: τοὺς μὲν ἀπλανεῖς ἀστέρας συνδεδέσθαι τῷ κρυστάλλῳ, τοὺς δὲ πλάνητας
ἀνεῖσθαι. The word πλάνητες cannot refer to the sun and moon alone; Empedocles had
special theories about them.
8 Ar §9: τοὺς δὲ κομήτας σύνοδον πλανήτων φλόγας ἀφιέντων. This sounds as though
he were thinking of a regular collision; in that case, Anaxagoras would have had to
station all the planets in the same region. The comet’s tail was explained as “flame,”
with the help of the analogy of flint. Aristotle (Mete. 342b25) speaks of σύμφασις,
“apparent conjunction”; but, since he includes Anaxagoras and Democritus together
(cf. below, n. 74), he may have obscured the difference between them.—Anaxagoras
was the first to make neighbors of the sun and moon (Eudemus fr. 147; cf. above, n.
$8).—It may remain an open question whether the sentence in Hebd. 2, lines 64ff, τὰ
τοίνυν ἄστρα τὰ οὐράνια ἑπτὰ ἐόντα τάξιν ἔχει τῆς τῶν ὡρέων ἐκδοχῆς, refers to the 7
planets, as Boll contended (NJb 1913 = KISchr 220ff), following the commentary of
ps.-Galen (CMG XI 2.1 p. 35). Kranz agreed with him (NGG 1938, 142, Kosmos 32
n. 8) as did Gundel (RE XX 2040) and Roscher (Hebd. p. 134 n. 191; but he rejects it
SBLpz 1919.5, 65ff). The dating of Hebd. depends precisely on the question whether it
shows knowledge of the planets and the spherical shape of the earth. In my opinion
the thesis that the plancts were known to the author falls with correct interpretation of
the phrase ὡρέων ἐκδοχή, which must mean “τῆς succession of the seasons”’ (cf. Hebd. 4,
14). The constellations named in the next paragraph are connected with this (Arcturus,
the Pleiades, the Hyades, Orion, and the Dog), but the 7 planets are not (except, of course,
301
ἦγ, ASTRONOMY AND PYTHACOREANISM
also in Democritus, He has been severely criticized for rejecting the
sphericity of the earth, and consequently his entire astronomical
system has been called backward and almost primitive.6? But a book
title like "Exmerdopara should make us prick up our cars; if Diels is
right about its meaning, the subject was a projection of the armillary
sphere on a plane surface, or in other words, mathematical astron-
omy.** Democritus was well informed in both mathematics and
astronomy and wrote a whole book περὶ τῶν πλανήτων.59 His series
of heavenly bodies, from the carth as center, was moon, Venus, sun,
planets, fixed stars; and the plancts themselves were put at various
distances from the earth.?° The reason for this arrangement is that in
the “whirl” the stars nearer the earth lose ground relatively to the
fixed stars. Thus the planets are integrated into the world system, with
different distances according to their different “speed;” Democritus
knows that there are a number of planets, and that they have definite
orbital periods, in some cases longer than those of the sun. The special
position of Venus points to influence from Babylon, where the trinity
of sun, moon, and Venus is attested very early.”! It is incredible that
Democritus should not have known the five familiar planets.
ee
the sun). Perhaps the words ἑπτὰ ἐόντα allude to the fact that certain constellations have
7 stars (the Bear, Pleiades, Orion, and, according to Pherecydes, the Hyades; FGrHist
3F90, Hippias DK 86B13); Varro in his Hebdomades (Gell. 3.10.2) mentioned the seven-
group of the Septentriones and the Pleiades even before the planets. (For seven-groups
of fixed stars in Babylon, see Meissner 407f.) Ζεὺς αἰγίοχος (Hebd. 6.2, line 17) cannot
mean the planet Jupiter (as Kranz thinks, NGG 1938, 125 n. 2; cf. Kosmos 32 n. 8), since
this manner of referring to the planets is only attested for a much later period (above,
ἢ. 6; Cumont’s conclusion could only be said to be “corrected”—Kranz, NGG 1938,
126-—if there were no other possible interpretation). There remains Boll’s idea that it
means the sky, because mention is made of its “change of color,” i.e. the alternation of
day-sky and night-sky (AbhMi 1918, 25; cf. below, ch. IV 3, nn. 27.64).
* The “childish character” of Democritus’ astronomy is spoken of by Burnet, EGP
339; 1.c., he was not a Pythagorean! (Cf. above, ἢ. 31).
$8 })K 68B11q; also the n. II 141, with ref. to Ptol. Geogr. 7.7.
* Fr. $b. In comparing Democritus with the allegedly more progressive “Pythagorean”
astronomy of Plato, one must bear in mind that for Plato we have detailed expositions in
his surviving books, but for Democritus only scattered and incomplete citations. If one
matches the doxographical accounts of Plato with those of Democritus, the latter comes
out fully as well (compare [/] Aét. 2.13.4/12, 2.15.3/4, 2.20.5/7, 2.25.6/9, 2.29.6/30.3);
an outstanding difference is that he gives physical causes rather than mathematical des-
criptions (cf. 2.16.1/7, 2.23.6/7). In his explanation of the Milky Way, Democritus
(unlike Aristotle) has the right answer (Aét. 3.1.6).
7 AB6, Ago §4; Lucr. 5.621ff = Democritus A838.
1 Shamash, Shin, Ishtar, represented as a group as early as the Naramsin stele; see
Cumont, AC 1935, 10 ἢ. 7; Meissner 18ff, 4o2ff. It is not entirely impossible that Par-
micnides is subject to oriental influence in the important position he gives to Venus (cf.
above, n. 40); but Democritus is certainly not following Parmenides. In fact, he differs
from Leucippus as to the position to be assigned the sun (above, n. 58).
312
1, The World and the Planetary System
Seneca, though, says that Democritus did not “yet” have an adequate
knowledge of the courses of the five planets. “ Democritus quoque . ..
suspicari 56 ait plures stellas esse, quae currant, sed nec numerum illarum
posuit nec nomina, nondum comprehensis quinque siderum cursi-
bus.”’?2 Yet this does not mean that Democritus “knew neither the
number nor the names of the planets.” According to Seneca’s report,
Democritus expressed the suspicion that there were “more’’ planets,
which surely means “more”’ in comparison to those he did know and
describe. Seneca, or his source, sees in this belief—in which Democritus
was correct, as we now know—a deficiency or shortcoming; the
astronomy of later antiquity thought it had attained final results,
and no longer reckoned with undiscovered stars.”
Eudemus must have found Democritus’ arrangement of the planets
incorrect, too, because of the special position of Venus. The “correct”
one was, however, included in the system of Philolaus; for the sequence
of the ten “divine bodies” was, in the unanimous testimony of
Aristotle and the doxographers,’ central fire, counter-earth, earth,
moon, sun, five planets, heaven of the fixed stars. If we consider
only the portion between earth and heaven, this is the order accepted
by Eudoxus, Plato, and Aristotle. In addition, this is the system
which Aristotle ascribes simply to “the Pythagoreans;” so nothing
scems in the way of the assumption that Eudemus meant the same
Pythagoreans, and his report that “the Pythagoreans”’ had established
the order of the planets referred precisely to the system of
Philolaus.
One would hardly suggest that Philolaus was dependent on Democ-
ritus;76 rather, there must have been a common source, someone who,
in the interval between Anaxagoras and the time of Philolaus and
Democritus, introduced to Greece, from Babylon, detailed knowledge
about the planets. In fact, this very epoch is that of the efflorescence
of professional Greek mathematics and astronomy. The floruit of
72 QNat 7.3.2 = Democritus Δ02.
73 Frank 202, followed by Gundel, RE XX 2040; more cautiously Cumont, AC 1935,
of: Did Democritus intentionally ignore the divine names of the planets?
74 See Zeller I 1107 ἢ. 2, and Heath, Aristarchus 128. Democritus explained comets
as a phenomenon of reflection between planets or a planet and a fixed star (Ag2, to which
Arrian ap. Stob. I p. 229.5 Wachsmuth and Posidonius ap. Schol. Arat. 1091 should be
added; cf. ἢ. 66).
78 Arist. fr. 203, Philolaus A16. Gundel (RE XX 2t1oof) erroneously takes the system
described at Plut. De an. procr. 1028b as that of Philolaus.
τὸ Apollodorus of Cyzicus made him Democritus’ teacher. Cf. above, ch. HI 2, ἢ. τοι.
313
IV. ASTRONOMY ANI PYTHAGOREANISM
Hippocrates of Chios was about 440. 8.0.2? It ἧς true that our only
direct information about his planetary theories concerns an exceptional
case, that of “the” comet: but in this is implied that he discussed the
behavior of the planets in detail, They move within the tropics, that
is, in the area of the ecliptic; and they “stay behind” the fixed stars,
each along his peculiar orbit (ὑπολελειμμένον ὅλον τὸν ἑαυτοῦ κύκλον,
Arist. Mete. 343a6), that is, they have definite periods. Heavenly and
terrestrial events affect cach other—the” comet sucks up moisture.
But the spherical plan of the celestial movements is already there,
in some detail: the parallel circles, oblique to the horizon and cut
by it at various angles.78 Hippocrates probably was influenced by his
countryman Oenopides, who had much to say about “circles” and
““inclinations,”’?9
June 27, 432 B.c. is the date of the summer solstice observed by
Meton, who was also “astronomer and geometer.”®° We know that
a DK 42; on his theory of comets, Arist. Mete. 342b36ff = DK 42.5 (Olympiodorus
on this passage, DK 42, speaks of the “sixth planet,” showing that he assumes a theory of
also Burkert, Philologus 1959, 193 n. 1).—Hippocrates is just once called Πυθαγορικός
in a late source (Schol. Arat. P. $46.21 Maass), but Aristotle distinguishes him from the
18 The way the problem is posed, whether the segment of the parallel circle above the
horizon or that below is greater (Arist. Mete. 343411, 18), corresponds to the method
followed for the quadrature of the lune (Eudemus fr. 140: below, ch. VI 1).
DK 41.2. He was ἃ somewhat younger contemporary of Anaxagoras, named to-
gether with Eudoxus by Hecataeus of Abdera, FGrHist 264F25 = Diod. 1.98, and also
CCAG VIII 3, p. 95.17 (a fuller version of the text given at DK 41.1a), where one reads,
πρῶτος... τὰς ἀστρολογικὰς μεθόδους ἐξήνεγκεν εἰς γραφήν. The date given, “end of the
Peloponnesian War,” is inexact; he is connected with Zeno, Herodotus, and Gorgias.
He was probably older than Meton. Cf. also von Fritz, RE XVII 22 58-2272. Schiaparelli
(Vorl. 15) interpreted his 59-year “great year” as comprehending the cycles of all the
planets (2 x Saturn, 5 X Jupiter, 31 x Mars, 59 X sun, Venus, and Mercury, 729 x
moon; the 9-year cycle of Saturn was known in Babylon: van der Waerden, Hermes
1952, 138.3, 139). Tannery (Astr. 143) also interprets Philolaus A22, which obviously is
connected with Oenopides, in the same way. But the tradition brings Oenopides’ “ great
year” with that of Meton (DK 41.9), and that has nothing to do with the planets and is
only concerned to coordinate the courses of the sun and moon. This is doubtless how
Ocnopides’ ‘great year” is to be interpreted (von Fritz, RE XVII 2262f). Oenopides is
never directly called a Pythagorean; there is a point of contact in the theory of the Milky
Way (below, n. 116), as well as a later charge of plagiarism (above, n. 37), and a remark-
able mélange of myth and science, though this is not necessarily a criterion of Pythago-
reanism.,
* Schol. Ar. Av. 997. On Meton, see Heath, Aristarchus 293-295; Kubitschek’s RE
article (XV 1458-1466) is unsatisfactory.
314
t. The World and the Planetary System
he had some contact with Babylon, however it was established, for
his nineteen-year cycle had been uscd there since 400." Meton is
named several times along with Eudoxus.®? He scems to have been
familiar not only with the idea, but with the graduation, of the
zodiac.®§ In the Babylonian manner, he designated the ascension of
various fixed stars as weather signs, and Democritus followed him
in this.* Tzetzes ascribes to Meton a doctrine of the “Great Year”
and of the destruction of the world when all the planets meet in the
sign of Aquarius.®> This may be merely a confusion of Meton’s
nineteen-year cycle with the World Year;8 and from a single rather
indefinite and general note we cannot tell for certain in what sensc
Meton dealt with the “distances” of the stars.87 The fact remains
that at the epoch of Meton and Hippocrates we can ascertain the
presence of not only empirical astronomical data borrowed from
Babylon, but also a precise, geometrical conception of a sphcrical
universe. From this time on, the general public was interested in
astronomical topics.8® This suggests that by then a fairly accurate
*'Van der Waerden, Anf. 112. There was one intercalation, which does not fit into
the system, in 386 B.c.
δὲ Columella 1, praef. 32, 9.14.12; Schol. Basileios Pp. 196.24-197.2 Pasquali.
88 Columella 9.14.12; van der Waerden, Anf. 170; cf. ch. IV 2.
δὲ Used in [Geminus] p. 212.3 Manit., Ptolemaeus, Phaseis. On the close relationship
of this literature with Babylon, see C. Bezold and F. Boll, ‘‘Reflexe astrologischer
Keilinschriften bei griechischen Schriftstellern,” SBHeid 1911.7; F. Boll and A. Rehm,
“Griechische Kalender, III,” SBHeid 1913.3; Democr. fr. 14.
$5 Chil. 10.534ff, 12.219ff, 12.283ff. The statement that Meton was the first to write a
book on astronomy (12.128, 214) has to compete with the same statement applied to
Oenopides (above, n. 79).
86 Cf. the confrontation Schol. Arat. Ρ. 478.5. But before Berossus (fr. 37 Schnabel ==
Sen. QNat 3.29.1 = FGrHist 680F21), not only did Plato know of the destruction of the
world by fire and water (Tim. 22c; the “great year” 39d), but Philolaus also did (A18):
Φιλόλαος διττὴν εἶναι τὴν φθορὰν τοῦ κόσμου, τὸ μὲν ἐξ οὐρανοῦ πυρὸς ῥυέντος, τὸ δὲ ἐξ
ὕδατος σεληνιακοῦ, περιστροφῇ τοῦ ἀέρος ἀποχυθέντος" καὶ τούτων εἶναι τὰς ἀναθυμιάσεις
τροφὰς τοῦ κόσμου. This comes under the title, πόθεν τρέφεται 6 κόσμος. The connection
of φθορὰ τοῦ κόσμου and τροφὴ τοῦ κόσμου can only be understood to mean that the
catastrophe happens to only a part of the world, doubtless the region of the earth (Zeller
I 549f). Parmenides and Heraclitus had taught that the stars are nourished by ἀναθυμιάσεις
(Aét. 2.17.4).—The catastrophe idea has nothing to do with Philolaus’ “great year,”
A22; for the latter takes a period of only 59 years. See further, on the “great year,”
van der Waerden, Hermes 1952, Anf. 116-119. There is much controversy over the
evidence of Heraclitus, frr. 66, 100, A13; cf. Reinhardt, Hermes 77 (1942) 1-27, 225-248;
Kirk, Heraclitus 300-305, 359-361; M. Marcovich, RE Supp. X 297-303.
87 Schol. Basileios ed. Pasquali (GGN 1010), 196.24~197.2.
88 Cf. Eur. fr. 861 (below, Ch. IV 2, n. 47), Gorg. Hel. 13, Pl. Gorg. 451c, Xen. Mem.
4.7.5. Here Socrates is said not to have wished to spend time on hairsplitting subtleties
about the movements of the planets or the like: this is of course a gibe at the Academy,
but, in the light of the other evidence, ought not to be considered an anachronisnn.
114
IV. ASTRONOMY ANN PYTHAGOREANISM
notion of the planets and their periods, as well as their assignment to
various gods, had become familiar to the Greeks.
Probably the historical situation had_ its importance. From 500
to 479 B.C., a state of war prevailed between the Persian Empire and
nearly all the Greeks and made peaceful contacts difficult, if not
completely impossible.*® This seems to have been the only period of
any considerable extent during which conncctions between Greece
and the Orient were almost completely severed. In this era of isolation
the most characteristic endeavors of the Greeks were reaching their
maturity; at its beginning, Parmenides was active. Then, as normal
conditions once more returned® and new relationships to the Orient
became possible, the Greeks were able, thanks to the progress made in
the meantime, to evaluate and select in a way quite different from that
of Anaximander’s time.
The opinion is widespread that Pythagoras himself, who is supposed
to have traveled in the East, brought this astronomical knowledge
back to Greece with him and passed it on through his school. In fact,
he is thought of as the most important link in the transmission of
oriental science to the Greeks. More cautious scholars are more likely
to speak not of Pythagoras, but of the early Pythagoreans, who are
supposed to be the only Greeks before Philolaus to have any advanced
astronomical knowledge.
This opinion leaves unanswered the question why we can detect no
influence of such knowledge, even in relation to the planets. In Par-
menides and Empedocles, at least, one would have expected to find
some traces; and if Oenopides “stole” the determination of the
obliquity of the ecliptic from Pythagoras, why did he not also take his
knowledge of the planets? Since there is no apparent reason why the
doxography should report the views of Democritus in more detail
than those of carlicr thinkers, we are driven to the assumption that the
planets did not really play so important a role with them. The Pythag-
oreans’ astronomic crudition would be, then, a secret doctrine with
no effect on others—a buried treasure. For Democritus’ astronomy
| " Theniistocles was probably not the only Greek who learned Persian in those days
Phe Democritus story represents Magi and Chaldaeans coming with Xerxes to Abdera
(DL, 9.34). On the other hand, Eudoxus was later prevented by political reasons from
traveling in Persia, and went instead—recommended by Agesilaus—to rebellious Egypt
(above, ἢ. 3). on
” As Democedes was active at the court of Darius I before the Persian Wars, so was
Ctesias under Darius IT and Artaxerxes II. .
*' Above, ch. II 2, nn. 15~16,
316
1. The World and the Planetary System
cannot in any case be derived from Pythagorcanism. There is not a
trace in the Pythagorean tradition of the special position of Venus in
his system, which points unmistakably toward Babylon. The con-
ventional view involves a remarkable double development; what the
Pythagoreans had long ago brought from Babylon had to be fetched,
a second time, from the same source.
But there is no good reason to assume a mysterious, secret pre-
Philolaus astronomy, belonging to Pythagoras or the Pythagoreans,
aside from one dubious inference from Parmenides and some even
more dubious late reports. Scholars have seized upon the assertion of
the “more genuine’ Pythagoreans, that the central fire is a force in
the interior of the earth, named “Eoria.® In fact, this epithet is applied
to the earth a few times in the fifth century,®* and Empedocles spoke of
fires beneath the earth.* Here, it is thought, we have traces of a geo-
centric system belonging to the early Pythagoreans, which we should
postulate anyway and which displays a suitable mixture of myth and
science. Nevertheless, the basis of the reconstruction, the report of the
“genuine Pythagoreans,’’ is an artificial reinterpretation of the reports
of Aristotle, and without independent value as a source. The only other
point is the name of ‘Eoria, but this is comprehensible as an expression
of the central location of the earth, without the idea of a central
fire. And since we may not regard every point of contact between
mythology and φυσιολογία as a priori Pythagorean—tradition even
associates Euripides and Anaxagoras as among those who called the
earth éoria®®—the conclusion that Pythagoreans are behind this
2 Above, ch. III 1, n. 72. The idea of a central fire within the earth was developed,
as the original Pythagorean theory, by H. Richardson (CQ 20 [1926] 113-133), following
Burnet, EGP 297f, then also by Wiersma, Mnemosyne 1942, 23ff. One piece of evidence
is Th. ar. 6.11ff (cf. above, ch. III 2, n. 139). A theory of fire in the earth is well attested
for the Stoic Archedemus (SVF III 264).
98 Soph. fr. 558 N. = 61§ Pearson, Eur. fr. 944; cf. Procl. In Eucl. 173.18f. At Pl.
Phdr. 246, too, ‘Earia is used of the earth (Dercyllides in Theo Sm. 200.7f, Macrob. Sat.
1.23.8). Cleanthes attacked Aristarchus of Samos as κινοῦντα τοῦ κόσμου τὴν ἑστίαν
(Plut. De fac. 9238).
Frr. §2, 62. This is natural enough for a Sicilian, with Etna before his eyes. At
Parmenides A37, a conjecture of Diels introduces the idea of fire beneath the earth; but,
uncertain as the whole world system of Parmenides is, this must remain in doubt (above,
n. 40).
"δ Anat. p. 30= Th. ar. 6.1§f . . . κατηκολουθηκέναι τοῖς Πυθαγορείοις of τε περὶ
᾿Εμπεδοκλέα καὶ Παρμενίδην καὶ σχεδὸν οἱ πλεῖστοι τῶν πάλαι σοφῶν φάμενοι τὴν
μοναδικὴν φύσιν ἑστίας τρόπον ἐν μέσῳ ἱδρῦσθαι... Here the subject of discussion is the
central position of the earth, in Parmenides and Empedocles, not a fiery center for the
earth, such as is attributed to the Pythagoreans. The hearth is the center of domestic life,
and the newborn child is carried around it (Αμφιδρόμια; cf. Nilsson I 95 n. 8; Hestia
μέσῳ οἴκῳ Hymn. Hom. Aphrod. 30). Fire burns on the hearth, not in it.
8 Anat. p. 30 (7h. ar. 6.186) = Eur. fr. 944 =: Anaxagoras A2ob.
317
IV. ASTRONOMY ANN PY THAGOREANISM
development is quite shaky. And there is no proof of special astro-
nomical knowledge, going back before Anaxagoras.
At the same time, the effort continues to attribute to the Pythagoreans
before Plato, and even before Philolaus, a different, geocentric planetary
system which became dominant in the later Hellenistic period. In
this, the sun is in the middle of the seven planets, flanked on each side
by three of them—Venus, Mercury, and the moon in the direction
of the central earth, and Mars, Jupiter, and Saturn in the direction of
the heaven of the fixed stars. This arrangement of the planets attained
a canonical position in astrology, and still determines the order of the
days of the week. It cannot be documented earlier than Archimedes, ””
yet it was attributed to Pythagoras and connected with a system of the
harmony of the spheres, in which the sun, both in its position and its
function, is μέση.38 This coincidence, so gratifying to every Pythago-
rean heart, and in general the “solar theology” that pervades the
theory, insured that the age and the originality of the system would
not go without defenders.9® An independent testimony seems to
corroborate this. Ptolemy says that the “older mathematicians”
(παλαιότεροι) put the sun in the middle, whereas “some later ones”
(ἔνιοι τῶν μετὰ ταῦτα) put all the planets above the sun. Since this is
preciscly the conception of Plato, Eudoxus, and Aristotle, the con- |
clusion is drawn that the “older” mathematicians in question are
pre-Eudoxan, and therefore Pythagorean, scientists.
* Archimedes: Macrob. Somn. Sc. 1.19.2, 2.3.13 (differently Hippol. Ref. 4.8 =
Archimedes ed. Heiberg ἢ 552ff; this is unreliable, cf. Boyancé Songe 61, REG 1952,
345f; Cumont, Lux 179 n. 1). Archimedes combined this with calculations of distance
that were criticized by Platonists: Macrob. Somn. Sc. 2.3.14 (cf. the mathematici, Cic.
Div. 2.91).—Hipparchus: Tannery, Astr. 127.—Nechepso-Petosiris (Plin. HN 2.88);
doubtless the source of its attribution to Egypt (Dio Cassius 37.19.2), and also the Chal-
dacans (i.e. astrologers; Macrob. Somn. Sc. 1.19.2).—Cic. Rep. 6.17, Philo V. Mos.
2.103, Quis rer. div. heres 224, Qu. in Exod. 2.75, Geminus 1.24ff, Cleomedes 1. 3, Manilius
L8riff (thus also, probably, Posidonius;: Boyancé, Songe 63, vs. Reinhardt, Kosmos 131f; _
differently Chrysippus, sce above, n. 7); Plin. HN 2.32. 41 {thus also Varro), Or. Chald.
cited Proc]. Ia ‘Vim. ΠῚ 63.22. Hultsch (RE II 1833) still wanted to refer this system to the
ancient Babylonians; Cumont (Lux 144) suggested “ Magusaioi” of the sth century B.c.
Hue there is no indication in the abundant original sources on Babylonian astronomy
that any such planetary sequence was known there (Boll, RE VII 2561ff; Neugebauer,
EnSe 168fF).
ἐκ Plin. HN 2.84, Cens. 13.3, Theo Sm. 1 38.0ff = Chalcid. 72f. On this interpretation
ofthe harmony of the spheres, sce Burkert, Philologus 1961; it is obviously dependent on
Eratosthenes.—This planetary system is contaminated with that of Philolaus by Plutarch,
De an. procr. 1028b (above, n. 75).
” Boyancé, Songe 78 ff.
Ptol. Syut. 9.1. The conclusion that Pythagoreans are meant: Boll, RH ΝῊ 2 $68.34;
Boyancé, Senge 62f; van der Waerden, Astr. 34ff, RE XXIII 2.1809.
318
1. The World and the Planetary System
The evidence of Eudemus makes considerable difficulty for this
view;!"! he can only be referring to the arrangement of the plancts
accepted by Aristotle, whose discovery he traces back to the Pythag-
oreans, for Simplicius is citing Eudemus merely to help explain
Aristotle. If there had been two different Pythagorean planetary
systems, Eudemus would surely have mentioned the fact; and he could
not have spoken of a discovery, but of a dilemma, since both systems
are worth about the same from a geocentric point of view. As it is,
there is not a trace of the second system, mentioned above, before
Archimedes 105 andit seems clearly to be later in origin. Beginning with
the naive idea of “‘sun, moon, and stars,” one’s first step in advance is
to recognize five of the stars as a special group—the planets—to
distinguish them from the fixed stars and class them, to a certain extent,
along with the sun and moon.! One sees the expression of such a
development in the systems of Philolaus, Plato, and Eudoxus: moon,
sun, five planets, fixed stars. Only when the close connection of the
“seven” has come to seem a matter of course, would one appreciate
the symmetry of the other arrangement, in which the insignificant
planet Saturn counterbalances the old familiar moon.
The conclusion drawn from Ptolemy does not hold up. When he
compares “old” and “newer” observations! he means by “old”
astronomers those from the third century on, as far as Hipparchus,
ca. 150 B.C. (who himself figures as the most important of the “‘an-
cients’’),°° by contrast with the “moderns” of the last hundred years
before Ptolemy. Eudoxus and Callippus are never cited in the Almagest ;
the astronomy of the fourth century has already sunk from sight.
Aristarchus, to be sure, assigned the sun to a central position; but
Plato’s great prestige kept his own system alive, even among. pro-
fessional astronomers.!%° The expression ““some later ones,”’ then,
refers to astronomers later than Hipparchus, who, doubtless because of
their orthodox Platonism, championed the older system. If even the
101 Above, ἢ. 45.
12 On Heraclides Ponticus, see above, ἢ. 7.
103 Above, n. 62.
104 Synt. 9.7 (II 262.11, 264.14 Heiberg), 9.10 (II 283.10), 10.9 (II 352.4), 11.3/4 (HI
386.17, 391.18), 11.7/8 (II 419.11, 425.6).
105 Synt. 3.1 (I 191.17); cf. 4.2 (I 270.1: of ἔτι παλαιότεροι = astronomers before
Hipparchus); 4.9 (I 328.5). The παλαιοί in the Apotelesmatica of Ptolemy are Nechepso-
Petosiris (RE XXHI 1837.49ff). Cf. also n. 97.
106. Hesitation of the μαθηματικοί as to the order of the planets: Aét. 2.15.5, Theo Sm.
p. 143.1 -- Chaleid. 73. The way in which Ptolemy presents the arguments for each
solution shows that the discussion was lively. Cf. also the criticism directed by certain
Platonic’ against Archimedes (Macrob. Sonu. Se. 2. 3.14).
319
ere 41 ΔΌΣΙΣ ANID PYTHAGOREANISM
brilliane Eudoxus had fallen inte oblivion, there can be very small
profit in looking to Ptolemy for evidence on pre-Platonic Pythago-
reanism,
Nor can internal indications of “solar theology” or the harmony
of the spheres prove that this late-attested system belonged to the
early Pythagorcans. Boyancé is able to cite a number of rather early
references to the identification of Apollo and the sun, and to the
connection between the sun and the harmony of the cosmos. But
the interrelation of myth and the interpretation of the natural world
is not Pythagorean alone; all Greek cosmology had to make its peace
with the existing myths; and the thought of cosmic harmony is by no
means tied to any particular planetary system.!°8 We shall not have to
alter the conclusion drawn from the passage of Eudemus: the arrange-
ment of the planets found in Plato and in the Philolaic system, and
no other, was known to Pythagoreans before Plato. We may conclude
that this went back to the acceptance of Babylonian information in
the time of Meton, just before that of Philolaus, and not to a more
ancient Pythagorean tradition.
The acusma which says that the planets are the “‘hounds of Per-
sephone”’ may well be older.°® Most nearly comparable to this is the
Babylonian designation of the planets as “‘rams.”!1° But we need not
suppose a direct dependence; the background is that of naive observa-
tion of the skies: the stars that catch attention by their independent
movements are thought of as living beings. These Pythagoreans,
therefore, are looking for the realm of Persephone in the skies; it is
also known that the sun and moon are the “Isles of the Blest.” As well
as this fits in, it does show that sun and moon werenot reckoned among
the planets. When one notices that certain stars “ wander about” in the
skics, the idea of mathematical arrangement and regularity does not
immediately spring to mind. There is here a kind of “astronomy, ἡ
107 Heraclitus fr. 100 (cf. also Anaximenes A14, Xenophanes A42, Kahn 1048), Oeno-
pides DK 41.7 (Helios = Apollo Loxias), Pl. Crat. 405 c-d, Scythinus DK 22C3.1 =
Plut. Pyth. or. 402a (the constellation Lyra that of Apollo, the sunray as plectrum; on the
date of Scythinus see FGrHist 1 3), Cleanthes SVF I 499, 502 (sunray as plectrum).
"In Hippoc. Vict. 4.89 we find ἡλίου δὲ ἡ μέση (περίοδος), namely between stars and
moon (also 1.10: πάντων ἐπικρατεῖται). The planets thus have no role here. On the
harmony of the spheres, see below, ch. IV 4.
™ Por. VP 41 (above, ch. II 4); Gundel, RE XX 2022. 36.
0“ Bibbu”; cf. Meissner 404. Also comparable is the designation of the planets as
“yenerals of Ahriman” in Iran; but it is not attested till late (A. Christensen, Die Iranier
(Munich, 1933) 229. Rougier (1ff) would like to show that in comparison with oriental
ideas the Pythagorean astronomy represented a definite forward step; but he overlooks
this acusma.
320
t. The World and the Planetary System
or we might better say, certain notions about the heavenly bodies
which prevailed among Pythagoreans and were peculiar to them—
for there is nothing clse in Greece like this—but they had nothing
to do with scientific Greek astronomy. There was a pre-scientific area
within Pythagoreanism itself. And this confirms that there was not a
continuous transmission of advanced astronomical knowledge from
the day of Pythagoras, but that scientific elements were added
only later on, in the course of the general development of Greek
science.
There is only a single passage in Aristotle to indicate that the Philo-
laic system was not the only one in vogue among the Pythagoreans,
stating that ““some’’ of them believed, as did Hippocrates of Chios and
his pupil Aeschylus, that “the” comet was a planet, a view also
shared by Diogenes of Apollonia? In the system of Philolaus, the
importance of the perfect number τὸ leaves no opportunity for any
such intrusive planet. It is difficult to interpret chronologically the
“similarity” of the Pythagorean theory to that of Hippocrates. The
latter gives a complicated explanation of the origin of the comet’s tail—
a phenomenon of reflection, he thinks, which happens only under
certain conditions, which is why the comet is so seldom visible.
The Pythagoreans do not go into the question of the tail, they only say
that this “planet” is only seldom visible and does not rise far above the
horizon.4 One might conjecture that the simpler theory is the older;
and in fact Aristotle mentions it first. But this makes it difficult to see
why the comet is not accounted for in the system of Philolaus. It
would also be possible that the Pythagoreans in question should be
dated later than Philolaus, and later than the refutation of Hippocrates
theory by the comet of 427/426 B.c. (above, n. 77). Their concern
was not to set up a bold new scientific hypothesis, but merely to
st a possibility.
es aes et Milky Way, too, Aristotle knows of different
theories held by Pythagoreans. Some see in it the path of a star that was
driven from its course in the catastrophe caused by Phaethon, others,
111 Arist. Mete. 342b30ff, followed by Aét. 3.2.1, ps.-Galen Phil. hist. 75, Schol. Arat.
p- 545 Maass. )
2 Ars: ἀστέρας εἶναι τοὺς κομήτας. "
118 The reflection of light is a favorite idea of the φυσικοί in the «(ἢ century B.c. Cf.
n. 11§ and below, ch. ΓΝ 3, n. 29. — .
114 The reference to Mercury at 342b32ff, being couched in direct discourse in the midst
of the indirect, is probably an explanatory note added by Aristotle.
321
ἃ route once followed by the sun. Philolaus (A18) spoke of fire
Howing from the sky. A doxographical entry attributes the second
answer to Ocnopides and adds that the sun changed its course out of
disgust at the meal of Thyestes.!'S The conclusion has been drawn from
this that Aristotle thought of Ocnopides as a Pythagorean;"7 but in
that case Democritus’ pupil Metrodorus of Chios, who held the same
view,"8 would also have to be a Pythagorean. A “good idea,” once
expressed, keeps getting repeated. It is entirely possible that Metrodorus
and the Pythagoreans were both dependent on Oenopides; we know
that Philolaus took over the idea of the “Great Year” from him.1!9
In any case, there is no trace, before Philolaus and Oenopides, of a
highly developed Pythagorean astronomy. |
2. THE THEORY OF PLANETARY MOVEMENTS
From Plato to Kepler, astronomy was dominated by the assumption
that all movements of celestial bodies were to be explained by the
combination of “perfect,” uniform, circular movements. On the basis
of this assumption there are two approximately correct explanations of
celestial movements, a heliocentric explanation like that of Aristarchus
or Copernicus, and a geocentric system like that of Ptolemy, with
epicycles and eccenters. From a mathematical or descriptive point
of view, the two systems are simply equivalent;} the decisive advance
$ 7 6 . «
was Kepler’s recognition that the orbits in question were ellipses
rather than circles.
5 Arist. Mete. 345a14ff, and Aét. 3.1.2 (DK $8B37c). The latter also includes under
the Jemma Πυθαγόρειοι the explanation of the Milky Way asa phenomenon of reflection
which goes back to Hippocrates (Arist. Mete. 345bIoff, DK 42.6); and he does the same
thing, at 3.2.1, with Hippocrates’ theory of comets. The error doubtless comes from the
meee ἘΠῚ a mathematician was automatically regarded as a Pythagorean (cf. above, n. 7.)
᾿ Ach. Is. §5.18 Maass = DK 41.10. Elsewhere it is stated that the stars reece
their courses because of the meal of Thyestes; i.e., they formerly rose in the west and t
in the east (Eur. El. 726ff, Or. toorff, Pl. Pol. 268e). | =
ἯΙ ἢ Fritz, RE XVII 2259; Boyancé, Songe 96, REG 1952, 348.
a pe : a τς 70A13: διά τὴν πάροδον (γίνεσθαι τὸν γαλαξίαν)" τοῦτον γὰρ εἶναι
9 Above, n. 79.
1See Tannery, Astr. 101; Neugebauer, ExSc 122ff, 183. Copernicus knew that his
system was no more adequate, descriptively, than that of epicycles and eccenters (ed
Soc. Cop. Thorunensis, Thorn 1873, 5.14), but he cited the “prima principia de ΕΘΝ
aequalitate,” which the epicycles violate. The “Copernican revolution” mainly affects
physics (aside from man’s feeling toward the world). Newton’s mechanics is Bibi onl
in a heliocentric system. In the modern theory of relativity, this difference ely een
geocentric and heliocentric has vanished again. See als
7 ; . oO J. Mittelstrass, Die
Phanomene (Berlin, 19 ae J Rettung der
322
2. Ene Eneory of Fianetary Movements
Eudoxus of Cnidus, taking his departure from this same assumption
of uniform circular motions, had devised a completely different solu-
tion,? and his system of concentric spheres, modified by Menaechmus
and Callippus, was adopted by Aristotle. It is hopelessly inferior to the
system of cpicycles and can be proven fallacious by many observations.
Eudoxus, however, was not just some crackpot theorist; he was one
of the most brilliant mathematicians of ancient times.* The fact that
he devised this system at all, and that it could hold its place for at
least a generation, is only comprehensible if the system of epicycles is a
later invention. We do not know its author, but in any case Apollonius
of Perga was using it about 200 B.c.5 As we can readily understand,
it drove the Eudoxan system so completcly from the field that the
latter is never mentioned by Ptolemy, and it is only the exigencies of
commenting on the works of Aristotle that have preserved the know-
ledge of it for us. Eudoxus’ solution was incomplete and was super-
seded, but, as the first attempt to explain the movements of the plancts
in a mathematical way, it was an astounding achievement. Eudemus
attests explicitly that Eudoxus was “the first’’ to apply himself to such
‘“hypotheses.’’¢
Thus both external attestation and internal evidence provide a
refutation, in advance, of all the attempts, ancient? and modern,® to
2 The standard study is Ὁ. V. Schiaparelli, Le sfere omocentriche di Eudosso, di Callippo
e di Aristotele (Milan, 1875; Ger. tr. by W. Horn: Leipzig, 1877). See esp. Heath, Aristarchus
1ooff and, for an attempt to reconstruct the method of Eudoxus himself, Becker, MD
8off. For Menaechmus and Callippus, see Dercyllides ap. Theo Sm. 201.25.
3 In the first place, the changes in size and brightness of the planets made it obvious
that there were changes in their distance from the earth, which would not be possible
according to Eudoxus’ system (cf. Arist. fr. 211, Sosigenes ap. Simpl. Cael. s05.10ff,
Heath, Aristarchus 208ff, Neugebauer, ExSc 154).
4 Eudoxus’ theory of proportion (Euclid, book 5) is highly esteemed by modern
mathematicians; see Hasse-Scholz 13ff; Becker, MD τομῇ; van der Waerden, SA
187-189.
5 Ptol. Synt. 12.1 = Apollonius fr. 59 Heiberg. Cf. Heath, Aristarchus 266f; van der
Waerden, SA 238; Neugebauer, ExSc 155.
§ Fr. 148. Sosigenes, ap. Simpl. Cael. 504.17ff, attests that until the time of Autolycus
(a little earlier than Euclid) no explanation had been found for the differing distances of
the planets. Cf. also Simpl. Cael. 32.16-27.
7 It seemed self-evident to the Platonist that Plato knew all the correct answers, at least
to any important question. See Dercyllides ap. Theo Sm. 201.7ff, Adrastus ibid. 188.25fF
(the only controversy was whether we should employ eccenters or epicycles; and Apol-
lonius had already shown that these amount to the same thing mathematically; above,
n. 4). Jamblichus, on the other hand (Procl. In Tim. HI 65.7), and Proclus following him
(In Tim. IN 96.27ff, 146.14ff) reject the complicated tangle of circles altogether; they
are only interested in the metaphysical answers, not in a mathematical description of
phenomena.
8 A recent champion of this view is van der Waerden, Astr. 371F; it is extended by
Becker, RhM 97 (1954) 89~92. The main text is Vim 38 ον the god created sun,
moon, and planets as “instruments of time’; ἔθηκεν εἰς τὰς περιφοράς, dy ἡ θατέρου
121
WV. ASTRONOMY AND PYTHAGOREANISM
find that the system of epicycley is alluded to as carly as Plato, and
therefore during the lifetime of Kudoxus.” Plato’s text can in fact be
understood without the epicycle hypothesis. To be sure, the ancient
Ln
περίοδος ἥειν, ἑπτὰ οὔσας ὄντα ἑπτά͵ σελήνην μὲν εἰς τὸν περὶ γῆν πρῶτον, ἥλιον δὲ
εἰς τὸν δεύτερον ὑπὲρ γῆς, ἑωσφόρον δὲ καὶ τὸν ἱερὸν ‘Eppod λεγόμενον εἰς [τὸν] τάχει
μὲν ἰσόδρομον ἡλίῳ κύκλον ἰόντας, τὴν δὲ ἐναντίαν εἰληχότας αὐτῷ δύναμιν, [τόν],
though securely attested, was deleted by Burnet. (MS y and older editions have τούς).
Taylor (Tim. 196) and Cornford (Tim. 105 n. 2) make ets govern an understood κύκλους,
and van der Waerden translates (Astr. 4 5) (the god put Venus and Mercury) “in
‘circles’ which themselves go in circles” (“in solche ‘Kreise,’ die einen Kreis gehen’’),
and calls this “ἃ very pregnant expression for epicycles” (46). Becker saw that the
expression is not at all pregnant—the decisive point, the introduction of new circles,
cannot depend on a word to be supplied!—and therefore corrected Plato’s text to read
εἰς [τὸν] τάχει μὲν ἰσόδρομον ἡλίῳ κύκλον, «κύκλους δὲ» ἰόντας τὴν [δὲ] ἐναντίαν εἰληχότας
αὐτῷ δύναμιν. The manuscript text, however, can be understood without alteration,
though there is a slight anacoluthon. Instead of the phrase we should expect, εἰς τὸν
τρίτον τε Kal τέταρτον κύκλον, attention is centered on the description of the peculiar
orbits of Venus and Mercury, using the two participles ἐόντας and εἰληχότας, and κύκλον
is taken ἀπὸ κοινοῦ both with εἰς τὸν and as inner object to ἰόντας. “(The god put) the
morning star and the one that is called sacred to Hermes into the circle that in speed is
equal to that of the sun—this is the circle they travel; but they are endowed with the
power contrary to the sun’s.” It is true that the puzzling ἐναντία δύναμις would have a
precise sense in the epicycle theory (the epicycle of the sun revolves to the right, that of
Venus and Mercury to the left; Adrastus ap. Theo Sm. 175.13ff, Chalc. τοῦ, van der
Waerden, Astr. 46; the phrase is discussed by Duhem 58f, Heath, Aristarchus 16 5ff,
Math. 1 311, Taylor, Tim. 196f). But in Eudoxus’ system, too, the fourth sphere of a
planet moves contrary to the third, whereas the sun has only 3 spheres. The expression
can also be understood in a quite general sense, like the phrase κατὰ τἀναντία ἀλλήλοις
ἰέναι, which was used of all the planets a little earlier (36d; Cornford, Tim. 8o0ff, 106ff).—
It is impossible to be sure that the sentence exhibits a terminological differentiation of
περιφορά and κύκλος as main circle and epicycle (as van der Waerden, Astr. 46); at 36c
κύκλος, φορά, and περιφορά are completely synonymous.—Another argument is made
from the “breadth” of the “whorls” in Rep. 616e (van der Waerden, Astr. 428 ). The
text and interpretation were very much debated in antiquity (Procl. In Remp. II 227ff).
Dercyllides seems to be the one who first interpreted the whorls in the light of the theory
of epicycles or eccenters (Theo Sm. 201.7f: cf. above, n. 7). Dercyllides was a Platonist
who wrote 11 books on Plato’s philosophy (Simpl. Phys. 247.30ff), after Hermodorus
and Eudemus (ibid.) but apparently earlier than Thrasyllus (Albinus 4, p. 149 Hermann;
sce RE V 242). The refutation of this version depends not so much on the fact that the
scries of widths “does not, to be sure, yet” agree (van der Waerden, Astr. 44), as on the
fact that the widest ring is that of the heaven of the fixed stars, which cannot have any-
thing to do with epicycles. The simplest and most obvious interpretation is still that in a.
tentative way the distances of the planets from each other are set to correspond to their
differences: the largest interval is that of the sphere of the fixed stars; Mars, Jupiter, and
Saturn form a group, as do Venus and Mercury. Jupiter and Saturn are most similar,
as can be seen, ¢.g., in Frank’s sketch (27). See also Tannery, Astr. 327, MSc VII 40ff;
Heath, Aristarchus 111, 156.
"Van der Waerden, Astr. 48 tries the expedient of supposing that the theory of epicycles
was first devised for the sun, Venus, and Mercury; but his own theory of the spindle
whorls will not fit this. And can we imagine that Eudoxus, instead of taking the natural
step of extending the theory of epicycles, proposed his own very different scheme,
applying it, against all common sense, to Venus and Mercury? In fact the ἱπποπέδη
(the figure-cight curve which is the product of Eudoxus’ theory) was surely devised to
explain the phenomena of “overtaking and being overtaken” observed in Venus and
Mercury. See also below, n. 61.
324
2. The Theory of Planetary Movements
tradition boldly asserted that Pythagoras himself was the inventor of
epicycles, or eccenters!’—one of the most horrendous examples of
anachronism in the construction of a science for Pythagoras. The
system of cpicycles was worked out by professional Greek astronomers
between Autolycus (fl. ca. 310 B.c.) and Apollonius of Perga and is
wholly irrelevant to the question of pre-Platonic Pythagoreanism."
In Plato’s Laws astronomy is endowed, more than any other science,
with a religious function. The study of astronomy does nor lead to
atheism, as was thought in the day of Anaxagoras, but is “dear to the
god” (821a). To be sure, its conclusions contradict popular opinion;
it is false that sun, moon, and stars “never follow the same path,
and the appellation “planets” is wrong: τὴν αὐτὴν yap αὐτῶν ὅδδὸν
é i 4 4
ἕκαστον καὶ ov πολλὰς ἀλλὰ μίαν ἀεὶ κύκλῳ διεξέρχεται, φαίνεται δὲ πολλὰς
10 Adrastus ap. Theo Sm. 150.12ff (cf. Chale. 77f): ἡ δὲ ποικίλη τῆς φορᾶς τῶν
πλανωμένων φαντασία γίνεται διὰ τὸ κατ᾽ ἰδίων τινῶν κύκλων καὶ ἐν ἰδίαις σφαίραις
ἐνδεδεμένα καὶ du’ ἐκείνων κινούμενα δοκεῖν ἡμῖν φέρεσθαι διὰ τῶν ζῳδίων, καθὰ πρῶτος
ἐνόησε Πυθαγόρας, τῇ κατὰ ταὐτὰ τεταγμένῃ ἁπλῇ καὶ ὁμαλῇ φορᾷ κατὰ ie
ἐπιγινομένης τινὸς ποικίλης καὶ ἀνωμάλου κινήσεως. The apparent variety mH 6 move-
ments of the planets results from the fact that, though in reality they are boun ᾿ aa
circular paths of their own and certain spheres of their own, they seem to us to fo ἊΝ a
course through the zodiac, as Pythagoras was the first to notice; thus to their =e "
simple, and uniform movement there is added, accidentally, a kind of variegate ae
nonuniform movement.” This unquestionably refers to the theory of epicycles an
eccenters (cf. Theo Sm. 152.2ff, 1§4.12ff); and it can be seen from the parallel ee
in Chalcidius that the whole thing, not merely the zodiac, is intended to be claime
for Pythagoras. Heath (Aristarchus sof) translates too loosely, so that the sales to
the epicycle system disappears and he is led to accept the passage as historica At ence.
—Procl. Hypotyp. 1.34: καὶ τοῖς κλεινοῖς Πυθαγορείοις, ὡς ἐκ τῆς ἱστορίας πάρει ἤφαμεν,
αἱ τῶν ἐκκέντρων καὶ ἐπικύκλων ὑποθέσεις ἤρεσκον ὡς ἁπλούστεραι τῶν ἄλλων iets
δεῖν yap ἐπ᾽ ἐκείνων καὶ αὐτὸν παρακελεύεσθαι τὸν Πυθαγόραν ζητεῖν ἐξ € ἐστον
καὶ ἁπλουστάτων ὑποθέσεων δεικνύναι τὰ ζητούμενα... Cf. Procl. In Remp. Il 230.2ff.
Iam. VP 31: δι᾽ ὃν (sc. Πυθαγόραν) περὶ... ἐκκεντροτήτων καὶ ἐπικύκλων νων ὀρθή τις
καὶ ἐοικυῖα τοῖς οὖσι παρεισῆλθεν ἔννοια... According to Simpl. Cael. ςο7.128,
Nicomachus “‘and after him, Iamblichus”’ attributed to Pythagoras only the theory of
eccenters. Geminus Is. 1.19ff only attributes to the Pythagoreans, directly, oe
postulate of uniform circular movement; but the proper solution is here too, o
theory of epicycles and eccenters.
Ve Ge acs “ἢ der Waerden (Astr. 37ff) traces the epicycle theory allegedly present
in Plato to Pythagoreans. “Pythagoreans . . . contemporaries of Plato or a little later,
was the verdict of Schiaparelli (Vorl. 60ff; similarly Duhem 433). Such are the expedients
used to rescue some bit of credibility for the testimonia about Pythagoras the scientist.
But after Plato and Archytas there was no longer any specifically ‘Pythagorean
astronomy, but only the general Greek professional astronomy. It may have been el
clides Ponticus who first postulated epicycles, for Venus and Mercury, though δὰ
interpretation of his fr. 109 is controversial (for epicycles: Schiaparelli, Vorl. 52ff; ;
Aristarchus 255ff, Duhem 4068, differently Frank 211f, followed by Wehrli and van der
Waerden, Astr. 62ff). Perhaps Heraclides referred to Pythagoras (above, ch. I 3, n. (1:
though it may not have been till after Hipparchus that Pythagoras was given credit for
this discovery (cf., on Aét. 4.13.9, below, ch. VI 1).
12 In later times it became a commonplace that the word “planet” is a misnomer. See
Plin. HN 2.12.
325
VW, ASTRONOMY AND PYTHAGOREANISM
φερόμενον εν ν (8224). T'wice more Plato returns τὸ this judgement, so
important and so satisfying to him, which, as he thought, refuted ἣν
mechanistic interpretation of the world and which makes it possible
to come, with λόγος, to the aid of the ancient Law” (890d). Of all
kinds of movement, the one most closely related to mind (voids) is
uniform, circular movement; it is perfect (898ab), and the heavenl
bodies move in conformity with it, not in any “crazy and disorder] ᾿
way (897d). “If they did not have souls, if they did not possess intelli
gence, they would never, in so exact a way, hold to such wondrous
calculations” (967b). Plato represents the Athenian as saying that h
heard this doctrine “neither as a young man nor very ae ago :
(8216), and we may apply this to Plato himself, for in the Republic
he was still of the opinion that real exactitude was impossible in the
physical world, so that the true astronomer should not depend on
sense perception but busy himself with purely ideal magnitudes and
movements.'3 In the Timaeus, too, Plato speaks without hesitation of
the “wandering” of the planets.!4 Thus it is easy to suppose that, onl
toward the end of his life, Plato learned and welcomed siti eanealls
a new theory of Planetary motion which attributed the apparent
irregularities to uniform circular movements.15
If we cast about for a planetary theory that Plato might have known
not only is the epicycle theory ineligible, but also the pera
of the earth, or any system like Copernicus’ .16 The only possibilit
aside from the system of Eudoxus, would be some kind of meen
18 Ἢ - “- ww > 3 . xy
; Bie $30a-b: ( ὁ τῷ «ὄντι ἀστρονομικὸς) οὐκ ἄτοπον, οἴει, ἡγήσεται τὸν νομίζοντα
γίγνεσσαι τε ταῦτα ἄει ὡσαύτως καὶ οὐδαμῇ οὐδὲν παραπλάττειν, σῶμά τε ἔ ἱ
ἜΤ ; Pa τε E€xoVTA καὶ
14 ,ὔ f Ἀ - 4
me a ; ible af ᾿ ἐπι φδῶὶ (394), τὰ τρεπόμενα καὶ πλάνην τοιαύτην ἴσχοντα
; aws (821d) the “normal” conception of the
job); Ὁ . planets is a βλασ iv.
panei e.g., noticed the difference between Timaeus and Laws (Cael ae oe
ἃ the diff ( . 480.5ff).
oy e pues ἐναντία δύναμις (Tim. 38d) and ἐπανακυκλήσεις Kai προχωρήσεις
" ἘΠῚ τὰ be applied to the Eudoxan system (above, ἢ. 8; Cornford, Tim. 136f )
‘i rank’s argument (202) is correct, up to this point. ies
᾿ oe = the earth was seen at this point by Schiaparelli, Vorl. 39ff, Burnet, EGP
ἫΝ 1: 3 para 2 aries others. Van der Waerden’s nominee is the Philolaic
sy: - 55). Dut neither of these systems explains the on ial
the πλάναι of the planets. Theref Fl τὶ ες ae
: ore Gruppe supposed that Plato knew
the helioce
system (KosmSyst 151ff; more cautiously Burnet, ThP! 348; C. Ritter SBHeid Ae
ee )s ἃ supposition neither provable nor likely. Certain other passages μανς
sen cited in this connection. The word ἐλλομέ i
Ἧ μένην at Tim. 400 was taken by Aristotle
“ ᾿ A τον ee : ἀμόνι following Heraclides, as Cherniss thinks Plato 4466)
a : 0 the revolution of the earth on its axis (cf. Ta ite
| 1 ι : . Taylor, Tim. 226ff; Heath,
bee 1748 ). The sal Beas solution is that of Cornford, that the earth turns
ὲ -ction Opposite to that of the fixed stars, thus canceli ie
mes it f the | ; celing out its movement and
appearing at rest (Tim. 120ff; Cherniss agrecs, Plato 554ff, with further elaboration:
326
ἡ. The Theory of Planetary Movements
postulate, completely regardless of observed phenomena, to be ascribed
to the Pythagoreans.
For to suggest, as Frank does, that the “discovery of the true move-
ments of the planets” was due to the Pythagoreans of Archytas’
circle,” is in itself absurd. This discovery is supposed to consist of the
thesis that the planets “describe geometrically perfect circular orbits
according to strictly mathematical laws” (28; cf. 201). What gave the
a An a - ὦ er a Ὁ τς .Ξ- ἐς
G. 5. Claghorn, Aristotle’s Critism of Plato’s Timaeus [The Hague, 1954] 74ff; van der
Waerden, Astr. $7; Kranz, RhM 1957, 114ff; cf. the ἀνελίττουσαι σφαῖραι of Aristotle).—
At Epin. 987b, the sphere of the fixed stars is called the true κόσμος, Os ἐνάντιος ἐκείνοις
σύμπασιν (the planets) πορεύεται, «οὐκ ἄγων τοὺς ἄλλους, ὧς γε ἀνθρώποις φαίνοιτ᾽
ἂν ὀλίγα τούτων εἰδόσιν. Whether one accepts Burnet’s conjecture (οὐκ) or with Taylor,
Tim. 169f, apprehends the passage as ironic, the passage would mean that the movement
of the fixed star firmament which carries the planets along with it, is rejected, 1.6. the
earth must move (Schiaparelli, Vorl. 4off; cf. J. Harward’s notes [Cambridge, 1928];
and Taylor, loc. cit.). This is supposed to mean rejection of Eudoxus’ system (characterizing
the most brilliant astronomer of the time asa “‘ person who knows little of these matters’’!).
But at 978dff, the movement of the heavens is accepted as self-evident (Cornford.
Tim. 91 n. 1; also Heath, Aristarchus 184f); the text is meant seriously, without «οὔκ».
The stress is on the word ἀνθρώποις; our human knowledge is basically inadequate; the
preceding context runs, βεβαίως δὴ διισχυρίζομαι τό ye τοσοῦτον, and the following,
ὅσα δὲ ἱκανῶς ἴσμεν, ἀνάγκη λέγειν καὶ λέγομεν' ἡ yap ὄντως οὖσα σοφία ταύτῃ πῃ φαίνεται.
_Critias 121¢: Zeus summons the gods εἰς τὴν τιμιωτάτην αὐτῶν οἴκησιν, ἣ δὴ κατὰ μέσον
παντὸς τοῦ κόσμου βεβηκυῖα καθορᾷ πάντα ..... here the text breaks off. Frank (207, 2174)
read this as an allusion to the central fire, and therefore to the system of Philolaus. Certainty
is not possible, since the sentence is incomplete (Cherniss, Plato 564). Clearchus fr. 8,
in a mythical context, seems to speak of a central Ἑστία outside the earth.—Theophr.
Phys. op. fr. 22, Dox. 494 = Plut. Quaest. Plat. 8.1.1006c (cf. Numa 11): τῷ Πλάτωνι
πρεσβυτέρῳ γενομένῳ μεταμέλειν ws ov προσήκουσαν ἀποδόντι τῇ γῇ τὴν μέσην χώραν
τοῦ παντός. Add Arist. Cael. 293a27ff: πολλοῖς δ᾽ ἂν καὶ ἑτέροις συνδόξειε μὴ δεῖν τῇ
γῇ τὴν τοῦ μέσου χώραν ἀποδιδόναι τὸ πιστὸν οὐκ ἐκ τῶν φαινομένων ἀθροῦσιν ἀλλὰ
μᾶλλον ἐκ τῶν λόγων: τῷ γὰρ τιμιωτάτῳ οἴονται προσήκειν τὴν τιμιωτάτην ὑπάρχειν
χώραν. Speusippus fr. 41 = Theophr. Met. 11822: σπάνιόν τι TO τίμιον ποιεῖ TO περὶ
τὴν τοῦ μέσου χώραν. Frank bases on this his thesis that the Philolaic system belongs to
the Old Academy (207ff); cf. Schiaparellt, Vorl. 36f; Burnet, ThPI 347f; Mondolfo,
Inf. 4518, van der Waerden, Astr. 55. The passage of Theophrastus is rejected as a mis-
understanding or arbitrary interpretation by Boeckh, KosmSyst 144ff; Tannery, MSc
IX 234; Cherniss, Plato 561ff; Cornford, Tim. 125ff, and others. It is certain that Theo-
phrastus is not referring to any of Plato’s writings, but to a δευτέρα φροντὶς which was
obviously made orally. Unless, like Cherniss, one entirely disbelieves in oral teaching by
Plato, the impression remains that the testimony of Aristotle and Theophrastus is mutually
corroborative, i.e. some members of the Old Academy defended the Philolaic system.
It is just as forced to read the doctrine of a central fire at the center of the earth (see ch.
Il 1, n. 92) into the passage of Aristotle (as Cornford does, Tim. 126ff) as it is to regard
his report as purely hypothetical (“if they were to proceed from a priori principles,”
Cherniss, Plato 560; against this is οἴονται at 293230; Mondolfo, Inf. 453). It would still
be possible that Theophrastus transferred to Plato an opinion that belonged to his pupils
(Heath, Aristarchus 18 46). Anyhow, all we learn about the Philolaic system is a terminus
ante quem (contra Frank 207, 278ff; van der Waerden, Astr. $5). Plato does not adopt a
system because it is new, but because, when he thought the matter over again (μεταμέλειν),
different aspects seemed important. Cf. the gloomy thoughts about the evil world soul
in Leg. ϑοός εἴ seq. and goéa.
17 Frank 26ff, 201ff; following him, Moreau, Ame 76f, Gundel, RE XX 2053.
327
oY oe ν τ EEE Ὁ 5 βἰπ Ὁ αι £F δ᾽ i PANIIRDAINISIM
planets their name, what drew the attention of observers to them at all,
was the fact that they do not, like the fixed stars, follow perfect,
uniform circular orbits, but tha, compared to the latter, they “wander
about.” It would also have seemed noteworthy that they do not, like
the sun, move regularly from west to cast through the zodiac (the
sun is not originally a “planct” but the paradigm of cosmic order,
ἡ τοῦ χρόνου τάξις).18 The first important lesson learned in Greece
about the planets, namely the identity of morning and evening star,19
means simply that Venus stands now to the right and now to the left
of the sun, moves away from it for a certain distance and then back,
overtakes the sun and is overtaken in its turn.20 The stoppings, the
retrograde movements, and the alterations in velocity were inherent in
the very discovery of the planets; they were known from the beginning.
A theory of the planets that takes no account of these matters explains
nothing at all.21
Now Plato makes it a matter of reproach against the Pythagoreans
that, at least in music theory, they value empirical data too highly,
and he surely gives no indication that the ideal astronomy demanded
in the Republic had actually been developed among the Pythagoreans.22
Aristotle says the Pythagoreans do not go beyond what is subject to
sense perception.” If this is so, it does not seem likely that they would
have set up a postulate that contradicted all the evidence of the senses,
without seeing any possibility of solution.
Geminus, on whom Proclus is obviously dependent, writes,
18 Cf. Anaximander fr. 1, Parmenides fr. 1.11f, Heraclitus fr.94: O. Gigon, Fondation
Hardt Entretiens 1 (Geneva, 1952) 137f.
19 Above, ch. IV I, n. 39.
20 Pl. Tim. 38d.
*! Frank finds that with their postulate of uniform movement the Pythagoreans had
“grasped the problem of the planets in its entirety, and had, in the essentials, also solved
it, but it does not seem that at that time they had yet explained all those pauses, retrograde
movements, alterations of velocity, and distortions of their orbital paths” (34f). In that
case, however, what have they explained?—Moreau (Ame 77) finds that the postulate of
circular motion is “le Principe méme de I’astronomie mathématique,” and suggests that
application to a specific system is not the main thing. But what is a principle that cannot
be applied?
ἐδ Rep. 5$30f; cf. ch. V τ.
3 Met. 989b209.
** Hypotyp. 1.34 (cited above, n. 10). Compare the following phrases (Geminus is given
first cach time): προέτειναν οὕτωϊπαρακελεύεσθαι τὸν Πυθαγόραν---ὑπέθεντο ἐγκυκλίους
καὶ ouadds .. . κινήσεις [ἐξ ἐλαχίστων καὶ ἁπλουστάτων κινήσεων---ἀποδοθείη τὰ
φαινόμενα δεικνύναι τὰ ζητούμενα---τὴν γὰρ τοιαύτην ἀταξίαν οὐ προσεδέξαντο πρὸς τὰ
θεῖα καὶ αἰώνια πρέπειν γοῦν τὰς τοιαύτας ( κινήσεις) τοῖς θείοις σώμασι μᾶλλον. When
Proclus adds to this, ὡς ἐκ τῆς ἱστορίας παρειλήφαμεν, he is not thinking of Eudemus
(as van der Waerden thinks, Astr. 48), but Geminus himself, whom Tannery long ago
recognized to be the most important source for Proclus in matters of the history of
mathematics.
328
2. The Theory of Planetary Movements
“The Pythagoreans were the first to approach such questions, and
they assumed that the motions of the sun, moon, and planets are
circular and uniform.” (Is. 1.19, tr. Heath). But the “stationary points
of the plancts are also explicitly mentioned, and in seas formulation
the problem of the Pythagoreans is, προέτειναν οὕτω, πῶς ἂν δι᾿ ἐγκυκλίων
καὶ ὁμαλῶν κινήσεων ἀποδοθείη τὰ φαινόμενα (Is, 1.2 1). Thus the Pythag-
oreans seck not arbitrary postulates, but the explanation of phenomena;
Geminus undoubtedly, like Adrastus and later writers, considers the
system of epicycles and eccenters to be Pythagorean.
Simplicius has a different account:?5
Eudoxus of Cnidus, as Eudemus recounts in the second book of his
History of Astronomy and as Sosigenes repeats on the authority of
Eudemus, is said to have been the first of the Greeks to deal with this
type of hypothesis.2¢ For Plato, Sosigenes says, set this problem for
students of astronomy: “ΒΥ the assumption of what uniform and
ordered motions can the apparent motions of the planets be
accounted for?”
Whether the statement about Plato forms part of the citation from
Eudemus and deserves any credence is a controversial question; the
anecdotal style and the fact that Simplicius names Sosigenes twice may
well rouse suspicion.2” But the sentence about Eudoxus, which is
certainly taken from Eudemus, is itself enough to refute the assertion
of Geminus. For what the latter attributes to the Pythagorcans—that
they “were the first to devote themselves to such inquiries and based
their account on uniform circular movements”—is attributed by
Eudemus, in almost the same words, to Eudoxus. He was “‘the first
to make use of such hypotheses,” namely “what uniform, orderly,
circular movements one must assume, in order to save the phenomena.
What doubt can there be that Geminus, in diverging from Aristotle’s
pupil Eudemus, is citing a late fiction, just as the epicycle theory he
cites is later in origin than the theory of Eudoxus? The explanation of
the movements of the planets, and thus the outstanding achievement of
mathematical astronomy, begins not with the Pythagoreans but with
Eudoxus. |
But it is new knowledge of which Plato speaks in the Laws, not a
25 Simpl. Cael. 488.18ff = Eudemus fr. 148, tr. Cohen and Drabkin. .
*° Namely (Simpl. Cael. 488.16), τίνων ὑποτεθέντων δι᾽ ὁμαλῶν καὶ τεταγμένων καὶ
ἐγκυκλίων κινήσεων δυνηθήσεται διασωθῆναι τὰ... φαινόμενα, :
μ The δον το about Plato was deleted by Hultsch (RE VI 939) as ἃ conjecture of
Sosigenes; it is defended by Friedlaender, Plato I 353 n. 15.
329
iV. ASTRONOMY AND PYTHAGORIANISM
postulate without a solution, Just as, in the Timacus, he mentions the
“overtaking and being overtaken,” the “ retrogradations and advances”
of the plancts,?8 he cannot, in his later book, regard them as cither non-
existent or unimportant; instead, he emphasizes the “exactitude”
with which the heavenly bodies follow the “calculations” (967b) ; this
is ““not easy to understand, but not extremely hard, either” (8216).
Since the matter is “proven,” it is to be incorporated into the educa-
tional system (822c). Ma@juara are “ prerequisite” to instruction in
astronomy (967e).2° The mathematical theory of the planets which
Plato knows in the Laws can therefore only be that of Eudoxus.3°
There is no chronological impediment to this conclusion;3! in content
the allusions in the Laws are compatible with Eudoxus’ system, ®2
and there is plenty of evidence for a relationship between Eudoxus
8 Above, ἢ. 14. Also, the word μιμήματα (Tim. 40d) obviously refers to a mechanical
model to illustrate the matter (Rivaud, Rev. hist. philos. 2 [1928] 4ff; Taylor, Tim 244f)
29 Cf. also Epin. g90a~c. ) )
30 That Plato was referring to Eudoxus seemed self-evident to Wilamowitz, Platon I
502; Bidez, Eos 150; W. Schadewaldt, “Eudoxos von Knidos und die Lehre vom un-
bewegten Beweger,” Satura: Festschr. Weinreich (Baden-Baden, 1952) I0$ ἢ. §; not to
Lasserre (181-182, 270).
3? On the chronology of Eudoxus, see Jacoby, Apollodor 314ff; von Fritz, Philologus
ὃς (1930) 478-481: G. de Santillana, Isis 32 (1940) 248-262; Merlan, Studies in Epicurus
and Aristotle (Wiesbaden, 1960) 98ff; Lasserre 137-139. Even with the generally accepted
later dating (ca. 395-342 B.c.), not only Leg. 821 but the Timaeus could well be referring
to Eudoxus (above, n. 8; see also O. Raith, Philologus 111 [1967] 33). Lasserre’s assertion
that “ein System kreisf6rmiger, regelmassiger Plantenbabnen nicht erst von Eudoxos. .
ausgearbeitet wurde”’ (270) is not based on any evidence and contradicts Eudemus. .
*? Plato’s statement that each planet traverses “the same path, not in many but in one
only” (822a) has been understood to exclude not only the ἕλιξ of Tim. 39a (the over-
lapping of daily movement and regression in relation to the zodiac) but also the system
of Eudoxus, in which 3 or 4 circular movements overlap (Burnet, ThP/ 347; Taylor
Tim. 210f, 231; van der Waerden, Astr. $5, and others; this is the reason for the efor
to prove that Plato believed in rotation of the earth or a heliocentric system; see above
n. 16). Of course the theory of epicycles and eccenters would be affected in just the
same way; and, since the Copernican system does not enter into the question, we would
have to renounce even speculating what Plato meant precisely (Taylor, Tim. 232). The
tradition of antiquity thought of both the epicycle system and that of Eudoxus as res-
ponsive to the Pythagorean postulate. Theo Smyrnaeus says (181.4ff), τὰ ἐν αὐταῖς
(ταῖς σφαίραις) ἄστρα τῇ τούτων ἁπλῇ καὶ ὁμαλῇ κινήσει φερόμενα κατὰ συμβεβηκὸς
αὐτὰ δοκεῖν συνθέτους καὶ ἀνωμάλους καὶ ποικίλας τινας ποιεῖσθαι φοράς. This evaluation
is of course correct. The naive conception which Plato rejects is characterized by formu-
lations like οὐδέποτε τὴν αὐτὴν ὁδὸν ἰέναι (8210), οὐδέποτε ἰόντας εἰς τὸν αὐτὸν δρόμον
ἀλλὰ πάντῃ πλανωμένους (8210), πλανᾶταί ποτε (8224), πολλὰς (ὁδοὺς διεξέρχεται) (8225).
This is the original idea, that the planets do not have fixed courses, but move at random:
they are “wandering” stars, “rams,” “hounds of Persephone.” In Eudoxus’ system
with the epicycles, a planet does not move faster or slower from time to time, nor ρος
times forward and sometimes back, but with a uniform velocity and direction, in a circle
which itself is moving, but just as regularly and in a perfectly circular course. Thus the
planet does have a specific, uniform path, governed by a uniform mathematical law and
repeating itself with mathematical regularity. This is what Plato means.
330
2. The Theory of Planetary Movements
and Plato. The astronomical conception which became the basis of
the world view of Plato and Aristotle and indeed dominated people’s
ideas of the world and their attitude to it until the time of Galileo—
the differentiation of the cternal and inalterable order of the heavens
from the chance and confusion of terrestrial events—comes not from
Pythagoreans but from Eudoxus. Plato did not, in a dogmatic spirit,
adopt a certain early Pythagorean system; he was au courant with the
scientific activity of his time, and was capable of appropriating its
most recent results for his own use.
Eudoxus is listed as a pupil of Archytas, in geomcetry,*4 and is
included by Diogenes Laertius among the Pythagoreans. With respect
to the indisputably original achievements for which Eudoxus became
famous in his lifetime,®> his supposed membership in the school has
little meaning. We know little about Archytas’ astronomy.* Circular
movement was for him the “natural movement,” which contains the
“ proportion of equality”’; this is why circles and round bodics occur
in nature—and this is applied, in the passage we have, to organic
nature.*” But Archytas was concerned with the analysis of curves
produced by a moving body,?* and it is tempting to suspect the in-
fluence of Archytas, the “inventor of mechanics” (D.L. 8.83), in the
celebration of the marvels of the circle in the Aristotelian Mechanics
(847b15-848a19), and the reduction of mechanical problems to the
relationships of larger and smaller circles.3® He had exerted an influence
on Plato’s thought as early as the Gorgias.4° Eudoxus went on from
there; but the admiration of the perfect circle takes us back to a much
earlier period. Alcmaeon spoke of the imperfect circle, of the failure
to join beginning and end which is the cause of a man’s death (fr. 2),
88 See D.L. 8.86ff; Jaeger, Arist. 15 (= 16f Eng. ed.); Lasserre 141-142, 254-256.
94 D.L. 8.86 Callimachus is cited here, though perhaps only for the fact that he studied
medicine with Philistion (see Pfeiffer on Callim. fr. 429).
δ D.L. 8.88 mentions a decree in his honor by his native city of Cnidus, and records
(8.91) that people called him “Ev8o€os.
86 There is a vague allusion in Hor. Carm. 1.28; Prop. 4.1.77. An isolated sentence of
Philoponus (Aet. mundi 522.20 Rabe) about the πέμπτον σῶμα τὸ bn’ ᾿Αριστοτέλους
καὶ ᾿Αρχύτου εἰσαγόμενον may be from Aristotle’s Archytcia (above, ch. HI, το. 94).
3? Archytas A23a = (Arist.) Pr. 915225: διὰ τί τὰ μόρια τῶν φυτῶν καὶ ζῷων, ὅσα
μὴ ὀργανικά, πάντα περιφερῆ;... πότερον, ὡς ᾿Αρχύτας ἔλεγε, διὰ τὸ ἐν τῇ κινήσει τῇ
φυσικῇ ἐνεῖναι τὴν τοῦ ἴσου ἀναλογίαν (κινεῖσθαι γὰρ ἀνάλογον πάντα͵), ταύτην δὲ μόνην εἰς
αὑτὴν ἀνακάμπτειν, ὥστε κύκλους ποιεῖν καὶ στρογγύλα, ὅταν ἐγγένηται; Cf. Frank 178
38 Above, ch. I 3, nn. 98-99.
°° F. Krafft in Antiquitas Graeco-Romana ac tempora nostra: Acta congr. intern. habiti
Brunae 1966 (Prague, 1968) 437-430; Dynamische und statische Betrachtunestveise in der
antiken Mechanik (Wicsbaden, 1970).
#0 Above, ch. 13, n. 156.
234
Vv. AOI RUINUM TY AND PY THAGOREANISM
as well as of the cternal circular movement of the divine stars (Α 12).
A Pythagorean acusma says that circle and sphere are the most beautiful
shapes.“! But even before Pythagoras, in Anaximander, the marvelous
properties of the circle keep the carth in equilibrium; the cycles of day
and year are even older ideas; and Homer himself speaks of the
“sacred circle.”4? Thus in the postulate of uniform circular movement,
which formed Eudoxus’ point of departure for the solution of his
problem, there is a reminiscence of more ancient speculations. Thinking
of the acusma and the evidence of Archytas, we may be justified in
seeing a Pythagorean inspiration here. But Eudemus’ testimony
precludes us from supposing that Pythagoreans earlier than Eudoxus
had succeeded in applying this concept to the apparent irregularities in
the planetary orbits and thus become the founders of mathematical
astronomy. Pre-scientific inspiration is in a sense the matrix of science,
but the two should be neither confused nor equated.
Except for certain stationary points and retrogradations of the
planets, the sun, the moon, and the planets seem to make the circuit of
the zodiac, each in its respective period, traveling from west to east.
This difference from the uniform east-to-west movement of the fixed
stars was interpreted in antiquity in two ways. Some of the older
natural philosophers spoke of the planets as “getting left behind”
(ὑπολείπεσθαι) by the fixed stars in the all-embracing cosmic revolution
—for example, Anaxagoras, Hippocrates of Chios, and Democritus.4?
Others thought of them as having a contrary movement of their own
from west to east, in spite of which they are carried along with the
general movement of the heavens, like ants crawling the wrong way
on a potter's wheel.“4 This theory of contrary movement is found in
Alcmacon,** Oenopides,** and Euripides.*” Plato rated this theory very
“' Above, ch. II 4, nn. 18, 23. Cf. Pl. Tim. 33b, Arist. Cael. 286bi0ff, Hypomn. 25,
Ocellus 1.15, etc.; also Empedocles fr. 27.4 == 28.2. |
Δ Il. 18.504. Cf. the circle of slabs around the shaft graves at Mycenae. Also, above
ch. 1 3, mn. 144-145.
“ Anaxagoras A78, Hippoc. DK 42.5 = Arist. Mete. 343a5ff, Democritus: Aét.
2.16.1 and A88 = Lucr. 5.621ff; Geminus Is. 12.148
“4 This comparison is made by Cleomedes (1.3) and others.
δ A4 Abt. 2.16.3. “Contrary movement” is given as the doctrine of the mathematic,
meaning professional astronomers, with the later addition, συνομολογεῖ καὶ ᾿Αλκμαίων
(above, ch. [V 1, n. 64).
332
2. The Theory of Planetary Movements
high and thought it was only the one that explained correctly the
velocities of the planets: Saturn is not the most rapid of them because
he is left behind least by the fixed stars, but the slowest, because he is
least able to free himself from their influence.**
The theory of contrary movement is regarded as specifically
Pythagorean and as a great advance over the Ionian theory. This is
inferred from Plato and Alcmaeon, whose evidence seems to take us
back to Pythagoras himself.*® But the relation of the two theories is
more complex.
Oenopides was an Ionian, a pupil of Anaxagoras;°° so that we hardly
ought to speak of a fundamental opposition between Pythagorean
and Ionian astronomy. And, if Euripides could allude in a play to the
‘contrary course” of the stars, we may be sure that the theory was
also known to Democritus; but he saw no occasion to adopt this
“advanced” view.
Actually there is no difference, from a purely descriptive point
of view, whether one speaks of “getting left behind” or of “contrary
movement,” as long as attention is confined to explaining the
apparent fact, that is the seeming displacement in the zodiac, which
is inclined with relation to the celestial equator. Hippocrates of Chios,®!
and thus doubtless the other advocates of the retardation theory, did
not maintain that this ὑπολείπεσθαι took place on parallel circles to the
celestial equator. No one can deny that the sun moves further to the
north in the summer and to the south in the winter. Naturally Anaxag-
oras and Democritus knew about the movement of the sun—and
the planets—in the zodiac.
The dicovery that was crucial for the description of the movements
of the planets was that of the zodiac. According to reliable tradition its
twelve signs were introduced to Greece by Cleostratus of Tenedos, a
pupil of Anaximander,®* after Anaximander himself had spoken of the
48 Tim. 38e, Leg. 8224 et seq.; cf. Rep. 617a, Epin. 987b.
49 Tannery, HScH 214; Burnet, EGP τιοῦ (Pythagoras himself); Heath, Aristarchus
50; van der Waerden, Astr. 27.
50 Above, ch. IV 1, ἢ. 79.
51 Arist. Mete. 343a8 (about the comet in Hippocrates’ theory): ὑπολείπεσθαι δ᾽ αὐτὸν
καὶ πρὸς ἄρκτον Kai πρὸς νότον... Geminus Is. 12.19 objects, against this theory, that
the ὑπολείπεσθαι would have to take place in parallel circles.
52 For the solstices there had to be a secondary cause, along with the vortex. Anaxagoras
found it in winds (A72), following Anaximander (A27) and Anaximenes (415). Demo-
critus’ answer to this problem is not clearly reported (4.80).
53 DK 6. The principal evidence is that of Plin. HN 2.31 = DK 6Bz2. See Kroll, RE
Supp. IV graf.
ἐγ. ASTRONOMY AND PYTHAGOREANISM
Oblique wheel of the sun. According. τὸ ἃ controversial report
Meton knew of the measurement of the ecliptic in degrees,55 |
The oldest Babylonian text yet known that refers to the “signs” of
the zodiac, not to constellations, is a horoscope from the year 410 B.c.56
Van der Wacrden takes expressions like “at the end of Pisces,’’ which
occur some decades carlier, as evidence for the introduction of the
twelve signs before that date, in place of the ancient names of con-
stellations.*” For even centuries earlier than this, people had marked
out the “path of the moon” in the sky and already discovered that the
sun, too, and the five planets, travel the same route.°8 In expressions
about the “path” of the sun or moon is reflected the conception that
is natural for the unsophisticated observer, that the sun and moon
and basically also the planets, make their way from constellation to
constellation in an eastward direction, One disregards, for the moment
the daily movement of the fixed stars—which is easy if one’s abe
vations are made at approximately the same time of night—and
discovers the peculiar movements of the other stars. They seem like
living creatures of a higher kind than ours, which move freely through
the heavens—whether one calls them “ rams,’ “hounds of Persephone,”
or gods. |
The conception of the independent movement of sun, moon, and
planets, their heavenly journey from west to east, is accordingly earlier
than the vortex theory of Anaxagoras and Democritus and is rooted in
naive observation and interpretation of the skies. Anaxagoras, Hip-
pocrates, and Democritus do not give less than this: what they want to
do is to contribute more. In place of unexplained, spontaneous move-
ment they posit a universal force, which works with the power of
necessity. As the cosmos is divested of its anthropomorphic features
the stars too must become things, governed by the ἀνάγκη ditiecis:
54 Anaximander As, 22. Cf Ta r i i
serge τ, ΤΕΥ, τὰς 96f. On the relation of this to the report
BS Columella 9.14.12: Hipparchus placed solstices and equinoxes in the first degree of
the appropriate zodiacal sign, but the author follows Eudoxus and Meton, who 2 them
in the eighth (“Eudoxi et Metonis antiquorumque fastus astrologorum”) Cf. pee
Eudoxos 222-224, though his citation of the Babylonians is out of date. Van dex Wacrden.
Anf,. 171, uses the Meton reference to establish a terminus ante quem for dating the
Babylonian system B.”—Eur. fr. 755 N., from the Hypsipyle, alludes to the zodiac.
ae ᾿ — J. of Cuneiform Stud. 6 (1952) 49; Neugebauer, ExSc 187; van der Waerden,
5” Anf. 124-125 (differently Neugebauer, ExSc 140); cf. 98-99, 132, on texts of th
6th century which still are concerned with stars rather than “sions.” Van der Waeden
dates the Babylonian “system A” of planetary calculation to the time of Darius I
(Archive for the Hist. of the Exact Sciences 5 [1968] 70-78; cf. Auf. 171: above. ἢ | )
58 The series Mul-Apin, van der Waerden, Auf. 77-79 ᾿ ἐῶ.
.
334
2, The Theory of Planetary Movements
The retardation of the planets answers the attempt to replace arbitrary
movement with movement according to natural necessity. ‘The
difference between the two theories, then, of retardation and of
contrary movement, comes to a difference between physical explana-
tion and mere description, whether naive or mathematical. The
transition from one to the other is not simply a forward step; rather,
the line of development turns back, on a higher level, to an carlier
stage. The rash conjectures of the Ionians on physical matters led into
a blind alley. The peculiarities of the planetary movements could not
be explained by the force of the cosmic whirl, and still, as people
were learning, the phenomena in question were regular and periodic.
The only way open to progress in astronomy was to abandon physical
explanations based on the necessary laws of movement and to adopt
purely mathematical description. The result was the Greek mathe-
matical theory of planetary motion, a tremendous achievement. It
was not possible, however, to find one’s way back from its complexities
to simple physical laws; so that, from Aristotle’s time on, the two-world
theory was dominant, regarding the realm of the heavens as wholly
different from and foreign to ours. Only with Galileo and Newton
did astronomy once more, from the heliocentric standpoint, align
itself with physics. Plato thought it was an inescapable conclusion
that the orderly movement of the stars is due to beings with souls;5"
it is a voluntary, chosen order. Here sophisticated Greck science harks
back to the pre-scientific way of thinking and comes to rest in it.
If Alcmaeon thought the stars were divine, he must have ascribed
to them a movement of their own; there is no advanced astronomical
theory involved here, as against Anaximander or Cleostratus—in
general it is obvious that Alcmaeon’s astronomical views are dependent
on the Ionians.6° Thus Alcmaeon’s statement is no evidence for
scientific advances attributable to Pythagoras. It was of course possible
for Eudoxus to take his departure from the idea of contrary move-
ment,® and, following out the principle in a mathematical way, to
lay the basis of the descriptive, though no longer physical, astronomy
of the Greeks.
58 P] Leg. 888e, 967a. Dercyllides ap. Theo Sm. 201.20 has the phrase κίνησις προαιρετική.
60 The sun is “‘flat’’: Anaximenes A7 84, Ar4-15; Alcmacon Aq. The moon is ‘ boat-
shaped”: Heraclitus A12, Antiphon fr. 28 (cf. Frank 188 n. 1); Alcmacon Aq.
61 In the Zodiac, there were to be seen two opposite circular movements, whose axes
are oblique to one another. Their overlapping produces the ἐλιξ of Pl. ‘Tint. 39 a-b.
Archytas may have developed the idea this far. Eudoxus may have wondered what the
result would be if the two movements were equally rapid, thus being led to his ἱπποπέδη.
335
μοι, ἡ. ἃ
Alexander has ἃ report, in the context of discussion of the harmony
of the spheres, about the Pythagoreans’ doctrine of the planets. This
is taken from Aristotle’s special book on the Pythagorcans. They
thought, he says, that the distances between the planets, and therefore
their velocities and the tones they make, are according to harmonic
ratios, and that those which move at the greatest distance move the
most rapidly (κινεῖσθαι δὲ τάχιστα μὲντὰ τὸ μέγιστον διάστημα κινούμενα) 6
Thus for these Pythagoreans Saturn is fastest, next to the sphere of the
fixed stars, and the moon js the slowest—the very view which Plato
ridicules as an absurd blunder. We cannot assume that Alexander has
made a mistake, for he repeats his statement with the notation,
WS προείρηκε (Ἀριστοτέλης). And in fact this conception fits the idea of
harmony of the spheres better than the one Plato favors; in the latter
the sphere of the fixed stars, which moves fastest, is followed by
Saturn, the “slowest” of the heavenly bodies, whereas in the former,
retardation provides for a progressive gradation—a necessity for the
connection with the musical scale.§* Far from simply taking over a
Pythagorean system, Plato hits the Pythagoreans, as well as Democritus,
with his criticism. There is one way out, as always: to posit a Pythag-
orean astronomy anterior to or foreign to Philolaus, and claim that
Plato is dependent on this; but there is no foundation for such an
assumption. In any case, the idea of contrary motion is not unanimously
held, even among the Pythagoreans.
Plato’s astronomy is not a copy of any Pythagorean system. To be
sure, the idea of the harmony of the spheres is Pythagorean, as is,
according to Eudemus, his planetary order, though here the only
difference from Democritus is in the position of Venus. There is
probably also a reflection of Pythagoreanism in his admiration of the
circle. Beyond this, Plato takes his place in the lively discussion of his
time, which could look back to almost a century and a half of Greek
astronomical speculation—an Ionian tradition, Anaximander to
Cleostratus and Anaxagoras to Oenopides. Plato takes a polemical
stance toward Anaxagoras and Democritus, because for him not
physical causality but logico-mathematical relationship is decisive.
The earth’s ability to hang free in space, and also the movement of
the planets, are comprehensible on grounds of mathematica] order,
62 Alex. Met. 40.7 = Arist. fr. 203; 41.5: κινεῖσθαι κατὰ ἀναλογίαν τῶν διαστημάτων,
ὡς προείρηκε. Cf. Arist. Cael. 290b21, on the music of the spheres, τὰς ταχυτῆτας ἐκ
τῶν ἀποστάσεων ἔχειν τοὺς τῶν συμφωνιῶν λόγους,
“Ὁ This is why Burnet attributes the “contrary movement” to Pythagoras himself (EGP
T10f), but the developed system of the harmony of the spheres to a later generation (306f).
Cf. below, ch. IV 4.
336
3. The Cosmos of Philolans
without any extraneous causation involved. Perhaps on this point
Plato learned from Archytas. With regard to planetary movement,
what he cared about was the mathematical theory, and this accomplish-
ment must be credited to Eudoxus.
3. THE COSMOS OF PHILOLAUS
Copernicus says that he got the stimulus for his revolutionary
cosmological system from ancient sources, and in this connection he
twice names Philolaus.1 And, since the time was far from past when
everything new was taken as a rediscovery of something ancient, the
Copernican system was current for a time under the name of astronomia
Pythagorica or Philolaica*—an error long ago laid to rest. Nevertheless,
the association with Copernicus still exerts a dominant influence in the
discussion of the Philolaus fragments; so subtly developed an astronom-
ical system cannot have been devised in the fifth century, it is
thought, and certainly not by the Philolaus who, as Plato said, provided
no “‘clear”’ rationale for his teachings. |
The doctrine in question, according to the consistent testimony of
Aristotle and the doxographers,* was that our earth is “one of the
1 Ed. Soc. Cop. Thorunensis (Thorn, 1873) 17, 6ff (“Philolaus Pythagoricus, mathe-
maticus non vulgaris’’), and in a passage deleted from the definitive seanigee ni : n.
See also the letter of dedication to Pope Paul III, 6.6ff. His sources were ] ay a
1.6. Aétius 3.13.1-3 (DK 44A21) and the report about “Nicetus,” i.e. Hicetas, in Cic.
. pt. 2.123 (DK $0.1).
sees ee ae 128f, Schiaparelli, Vorl. 17ff. .
ὅ Μὴ der Waerden, Astr. 49ff, 54. Before him Frank, especially, had sought to prove
the impossibility of the system of Philolaus in the sth century B.c. (3 ς, tie ies sip
followed by Rehm-Vogel (47), Gundel (RE XX 2056f), and others. ie : oe yne
1942, 25ff, tried to prove that Philolaus did not advocate this system 0 γῆι ο _
(above, ch. HI 2, n. 33). Its origin is attributed conjecturally to Hicetas (1 $0) by
nd van der Waerden.
ie eae Cael 293a18ff, 986a8ff, frr. 203, 204; Aét. 2.29.4 = Arist. fr. ae Walzer; also
Aét. 2.7.7, 3.11.3, 3.13.1 = Philolaus A16, 17, 21. On the relation of the “I ale el
of Aristotle to Philolaus, above, ch. III 1. Aristotle and Aétius agree not only a sub-
stance but in actual wording, so that it is not only permissible but mandatory seta
the material included in one line of tradition but not the other into the tota ee
(The “plus” in Aristotle: the creation of night and day, Cael. 293222; the arabs Ἢ
of distances and velocities, fr. 203, Cael. 290b21; the absence of parallaxes, Cael. Κῶ δεν :
the counter-earth and lunar eclipses, Cael. 293b23, Aét. 2.29.4. The “plus” in the uilo-
laus doxography: fire as περιέχον, A16; the sun as a burning-glass, Aig; the moon is
habited, A20; movement of the earth in the same direction as sun and moon, im an
oblique circle; world catastrophes, A.18. Mondolfo, Inf. 286.2, 34off, secs a oe
the fact that in Philolaus the world is surrounded by fire, but in Aristotle ee ry
ἄπειρον, κενόν [above, ch. I 2]; but this is a contradiction only if, like ἐν Ὁ Ὁ, Μὲ
supposes that the surrounding fire is infinite in extent, and there is no good reason to
do so.)
wr fF πὴ SEWN SOSA YES FPEILMAGURELANISM
stars” and along with the moon, the sun, five plancts, and an invisible
counter-carth” revolved about a “central fire.” The earth a planct!
This sccms to anticipate Copernicus’ momentous discovery, and one
involuntarily regards the Philolaic system as an attempt to explain, in
as clear a way as possible, certain specific astronomical abseevadione:
Nevertheless, this interpretation leads from one difficulty to another
The system of Philolaus would be truly Copernican if it could δέου
an explanation for the apparent irregularities in the orbits of the
planets. Frank thinks it does: “In this ‘Pythagorean’ or ‘Philolaic’
system ...the apparent retrograde movements and pauses find their
explanation ...in quite the same way as in our modern heliocentric
system.’’> For this reason, he thinks, the system must be later than
Eudoxus and Heraclides Ponticus. The fact remains, however, that it
was not the sun which the Pythagoreans thought occupied the central
position, but the central fire: and this, like the counter-earth. is always
invisible to us “because the earth is in the way. It liga that the
earth's circular orbit and its rotation on its own axis must be combined
in such a way that the earth always exposes to the central fire the side
Opposite the one on which we live. This is inconsistent with the main
point of the system of Aristarchus or Copernicus; for in the latter
the earth’s rotation on its axis explains the daily rising and setting of ie
stars, and its annual revolution about the sun explains the changing
seasons and the convolutions of the planetary orbits. If these two move-
ments are inseparably connected with each other in the system of
Philolaus, they can only explain one astronomical fact. Aristotle says
explicitly that the earth “by its circular movement about the center
creates day and night” (Cael. 293a22fF ). Thus the period of revolution
about the central fire, but relatively to the sun, is one day. This ex-
cludes the “Copernican” idea of explaining the distortions in the
orbits of the planets as distortions of perspective due to the movement
of the earth. The planets would have to turn retrograde in the course
of a single night—unless there are in fact no appreciable parallaxes!
This is the point of view taken by the Pythagoreans, according to
Aristotle. Even on the geocentric hypothesis, they thought, the
astronomer is not making his observation from the center, but is one
Ἵ Frank 38f, cf. 35ff, 2076. Similarly Gundel, RE XX 20 46: “their effort is to solve the
(planetary) problems that arise in a mathematical way.” More cautiously, Rehm-Vogel
47. Naturally those who are experts in the subject matter, like Martin, Heath Schiaparelli,
and van der Waerden do not commit this error.
5 Invisibility of the counter-earth because of the ὁ }
€ ἐπιπρόσθησις τῆς γῆς Arist. C
293b22, Simpl. Cael. 511. 34f, quoting Arist. fr. 204. δι ἐμὰ ἐν δον
338
4. The Cosnioa of Philolaus
earth radius away fromait; “there is no reason not to suppose, they
think, that celestial phenomena are the same, even though we are not
at the center,” bat on the earth which revolves about the central
fire (Cael. 293b25ff). In other words, this Pythagorean system, which
expressly denics a parallax as the result of the carth’s movement,
cannot provide any thcory of planetary movement and has no intention
of doing so. In this respect it is to be classified with pre-Eudoxan
astronomy, in which the capricious prancings of the plancts were
simply taken to be inexplicable. Frank does not hesitate to assert that
the central fire and the daily revolution of the earth around it are
“speculative and mythical reinterpretations’’ of a truly Copernican
system, made by the “philosophers of the Academy,” and he
therefore explains the traditional account as the corruption of an
unattested “Pythagorean system of scientific astronomy, 7 thus himself
abandoning the realm of historical reality in favor of arbitrary
speculation.
The effect of the earth’s movement in Philolaus’ system is the same
as that of simple axial rotation. One might see in this a magnificently
bold and fruitful idea: the daily rising and setting of sun, moon, and
stars, central facts of human life, become illusion, caused by an un-
observed movement of the earth, though it always must appear to be
fixed and at rest—a triumph of thought over mere appearance. The
earth moves each day, from west to cast, “in the same way’ as the
sun and moon,® in an “oblique circle.” Its path is therefore in the plane
of the equator, that of the planets in that of the ecliptic. The planets,
including the sun and moon, likewise move from west to cast, but
much more slowly; their angular velocity is less in proportion to their
distance from the central fire.® If one can forget about the unexplained
irregularities in the courses of the planets, this makes a system of
impressive symmetry. Each celestial body has one and only one circular
7 207, 208.1. For the central fire Frank substitutes “den (ideellen) Mittelpunkt des
ganzen Planetensystems”’ (35; cf. 207), about which the earth circles “in cinem Jahr?”
(208). This last item is taken over by Rehm-Vogel (47), but without Frank’s question
mark. Frank’s citation of Heraclides misses its mark (below, n. 19; Wehrli 97), and his
reconstruction is unsatisfactory astronomically. As long as the course of the sun remains
outside that of the earth (see Frank’s drawing, p. 36), there can be no adequate explanation
of the phenomena. On the “ideal center” Heath says (Aristarchus 278), speaking of Hera-
clides, “It is inadmissible to suppose that, in Heraclides’ time, any one could have assumed
that the place in the centre of the universe was occupied by nothing and that both the
sun and the earth revolved about an ideal point.”
8 ὁμοιοτρόπως, Philolaus A21. Elsewhere the expression “oblique circle” is always
used of the ecliptic (Andrissi 7f).
® Schiaparelli, Vorl. 14; Martin, Philolaus 1§5.
339
er ft CON SAUNT U MEE δ FEES δεῖ ΙΑ ΚΙ ΙΓ
course to complete! ~ provided, that is, that the sphere of the fixed
stars is stationary. But this very proviso cannot be allowed, according
to the explicit testimony of the sources! ‘Ten bodies “dance a rounde-
lay” about the sky;" nothing is stationary but the central fire, the ἑστία
in their midst. However the movement of the fixed stars is imagined,
they must carry the rest of the stars with them, in order to conform to
observed appearances.!2 Thus the astronomical significance and
special advantages of the Philolaic system have vanished. None of the
attempts to devise a precise interpretation of the movement of the
heaven of the fixed stars is successful.13 As a matter of fact the confusion
becomes complete only when we include the testimony of Alexander,
so often overlooked, that the stars move with a velocity proportional
to their distances from each other—the further away, the faster they
move.'* This means that the sphere of the fixed stars is the fastest of
all, and there is no difference from the “whirl” of Democritus. There
is no point in discussing the various possibilities of this system; one
can only analyze motion in relation to something stable; and the point
of reference in the system of Philolaus is the eternally invisible central
fire.> Proceeding in this fashion, one can conjecturally attribute any
kind of motions to the heavenly bodies, so long as they all participate
10 This is why it is brought into connection with Pl. Leg. 821f (above, ch. IV 2, n. 32).
1) Arist. Met. 986a10; τὰ φερόμενα κατὰ τὸν οὐρανὸν δέκα μὲν εἶναί φασιν. Arist. fr.
203 = Alex. Met. 41.3: τὰ δέκα τὰ κινούμενα σώματα, cf. 38.23; Philolaus Ar6: περὶ
τοῦτο δέκα σώματα θεῖα χορεύειν.
15 ΤΕ the fixed stars moved around the axis of the ecliptic, the constellations would
change position relatively to the poles and the equator; the polestar would not remain
such. If the fixed stars had a course around the pole and the planets did not, the courses of
the planets would be altered, with respect to the constellations, and the zodiac would no
longer be the zone of planetary movement.
18 Boeckh (118) thought of the precession of the equinoxes, and this was taken up by
Gomperz (GrD I 93 = I 114 and 544 Eng. ed.), though Boeckh later vacillated (KosmSyst
93). Knowledge of the precession is not attested before Hipparchus, and is not likely.
Van der Waerden, Astr. 54, thinks of a rotation of the sphere of the fixed stars through the
ecliptic in a period of 184 years, which is supposed to explain the retardation of the nodes
of the moon. But the author of the system does not seem to have any exact knowledge
about lunar eclipses (below, n. 31, and above, n. 12). Schiaparelli cut the Gordian knot
by simply rejecting the evidence and postulating “absolute motionlessness of the heaven
of the fixed stars” (Vorl. 11, 14f). Tannery, too, says (HScH 246) “‘l’essence méme du
systéme est Pimmobilité de la sphére des fixes.” See also G. B. Burch, Osiris 11 (1954)
267-294, esp. 282-285.
14 Above, ch. IV 2, n. 62.
1° According to the Pythagoreans, the motive power of the cosmos is localized in the
central fire: Simpl. Phys. 1354.2, 1355.3 (probably from Alexander, and hence ultimately
Aristotle).
340
3. The Cosmos of Philolaus
in them in such a way that their relative displacements conform to
observed facts;! but this is not what is usually meant by astronomy.
Things are different with the astronomical doctrines ascribed to
Hicetas, Ecphantus, and Heraclides Ponticus. Though there is con-
troversy about many details!” this much is clear, that Hicctas and
Heraclides believed that the outer heaven was at rest and emphasized
that from this hypothesis the phenomena could be explained just as
well as if it were thought of as rotating.1® Heraclides stated explicitly
that the apparent position of a planet with relation to the heaven was
determined by the straight line from the earth, that is, the eye of the
observer, to the planet.1? Thus he was dealing with the basic ideas of
perspective and of projective geometry, the differentiation of true and
apparent position, or true and apparent movement—in other words,
this is mathematical astronomy. It is relevant here that Philip of
16 On the idea of an unnoticeable, undetectable motion, cf. Martin, Philolaus 140ff,
1455, who refers to Ptol. Synt. 1.7 p. 24 Heiberg; Schol. Arat. p. 91.27 Maass; Schiaparelli,
Vorl. 13.18. The end result is about the same in the more complicated solution of Andrissi
(who is followed by Timpanaro Cardini 1946, 325). He posits two motions of the earth.
which run counter to each other and cancel each other out in their astronomical effect,
h. IV 2, n. 16, on Tim. 4ob). .
ae eg ae (According to Heath, Aristarchus 189, he is to be identified with
Hicetas of Syracuse, the tyrant of Leontini, friend of Dion and rival of Dionysius II.
Cf. Plut. Dion §8, Timoleon 1-32.) Ecphantus, DK §1. Heraclides, frr. 104-110. We
cannot enter into the controversies about the astronomy of Heraclides (sce above, ch.
IV 2, n. 11). In Tannery’s view (MSc VII 249ff, IX 2328, 2538) Hicetas and Eephantuy
are both characters in a dialogue of Heraclides (agreeing: Frank 138 and nn. qoaff
disagreeing: DK I 441 n; Mondolfo, ZM 349; van der Waerden, Astr. § Ἢ cf, γόλος
96). The conjecture is based on the naming of Ecphantus along with Heraclides in a
3.12.3 (= DK 51.5 = Heraclides fr. 104) and the similarity of some kind of ae
theory (DK $1.1-4, Heraclides frr. 118-121). Theophrastus reference to Hicetas (I oH
op. fr. 18 = DK $0.1) is not a positive disproof of this (contra DK;; ef. above, ch. I 2):
but the counter-earth, advocated by Hicetas, is never mentioned by Heraclides.—The
system of Hicetas, as described by Cicero, is impossible. Not only the fixed stars but also
sun and moon are to be stationary (DK 50.1). Tannery thinks Cicero made a mistake
(MSc IX 234f), as does Duhem (23); Frank thinks Hicetas exaggerated for didactic
purposes (n. 402); van der Waerden tries for a solution by slightly altcring the translation
(55). See also Wehrli, Herakleides 95. . .
18 Hicetas, DK 50.1: “‘eadem effici omnia quae si stante terra caclum moveretur :
Heraclides fr. 108: σῴζεσθαι τὰ φαινόμενα τοῦ μὲν οὐρανοῦ καὶ τῶν ἄστρων ἠρεμούντων,
τῆς δὲ γῆς... κινουμένης. Rotation of the earth on its axis is attested for Ecphantus and
Heraclides (DK 51.5, Heraclides fr. 104-108); on the connection with Arist. Cael. 293b30ff,
296a26ff, above, ch. IV 2, n. 16——Schiaparelli, Vorl. sof, and van der Wacrden, Astr.
58f, are wrong in concluding from Schol. Coisl. Arist. p. $0543 that the more genuine
Pythagoreans taught that the earth rotates, the sentence in question is a citation of Arist.
Cael. 293422; cf. above, ch. II 1, n. 74. . .
19 Fr, 109: “... solem οἵ lunam et luciferum ct omnes planctas, ubi coruny quisque sit,
una linea a puncto terrae per punctum steilac excunte demonstrari . . .
341
IV. ASTRONOMY AND PYTIAGOREANISM
Opus, who was mainly interested in astronomy, wrote a book en-
titled ᾽Οπτικά.39
There is nothing of this sort in Philolaus, only an invisible central
fire, an equally invisible counter-carth, unobservable movements
of the earth and the stars—mythology in scientific clothing, rather than
an effort, in accord with scientific method, to “‘save the phenomena.”
This is just what Aristotle says of the Pythagoreans: “not seeking
accounts and reasons to expain the phenomena, but forcing the
phenomena and trying to fit them into arguments and opinions of
their own.’’! The system of Philolaus is not a scientific astronomy,”
and there is no call to set it late chronologically on grounds of its
sophisticated and advanced nature.
There are details among the astronomical teachings of Philolaus that
point toward the fifth century, and in particular his remarkable theory
of the sun. The sun is a “glass-like” body which receives light and
warmth from the “‘ethereal fire” which it then “strains through”
certain “‘narrow interstices,’ so as to shine on the earth.?? His basic
20 Suda s.v. φιλόσοφος.
21 Cael. 293225, tr. Guthrie. Speculation and concentration on the world apprehended
by the senses are two interpenetrating tendencies in pre-Socratic philosophy. It was only
after Plato had made clear the dichotomy between immaterial and material that mathe-
matics, become independent, could set about from a new basis to “save the phenomena.”
22 Wilamowitz, Platon Il 93: “Ich fiirchte, es ist von Astronomie dabei wenig zu
ruhmen.”
23 A&t. 2.20.12 = DK 44Ar9. Also Ach. Is. 19, a passage that Diels did not include in
the Vorsokratiker because he thought Achilles was directly dependent on ps.-Plutarch
(Dox. 22ff). This thesis is not tenable, and it was withdrawn (orally) by Diels himself;
see Pasquali, GGN 1910, 221f. Achilles draws from a more detailed source, as the follow-
ing parallels show:
Aét. 2.20.12 (from Stob. and Plut.,
supplementing each other; DK 44410):
e - 4 -
ὑαλοειδῆ τὸν ἥλιον, δεχόμενον μὲν
-. + ~ ?
τοῦ ἐν τῷ κόσμῳ πυρὸς THY ἀνταύγειαν,
διηθοῦντα δὲ πρὸς ἡμᾶς τό τε φῶς καὶ
τὴν ἀλέαν,
ὥστε τρόπον τινὰ διττοὺς ἡλίους
γίνεσθαι
τό τε ἐν τῷ οὐρανῷ πυρῶδες καὶ τὸ
ἀπ᾽ αὐτοῦ πυροειδὲς κατὰ τὸ ἐσοπτροει-
δές,
εἰ μή τις καὶ τρίτον λέξει τὴν ἀπὸ τοῦ
ἐνόπτρου κατ᾽ ἀνάκλασιν διασπειρομένην
πρὸς ἡμᾶς αὐγήν. καὶ γὰρ ταύτην
προσονομάζομεν ἥλιον οἱονεὶ εἴδωλον
εἰδώλου.
342
Ach. Is. 19 p. 46.13ff Maass:
4 τ 3 ~
(τὸν ἥλιον) τὸ πυρῶδες καὶ διαυγὲς
é ~
λαμβάνοντα ἄνωθεν ἀπὸ τοῦ αἰθερίου
πυρός
4 ε ~ fa + , a fd
πρὸς ἡμᾶς πέμπειν THY αὐγὴν διά τινων
3
ἀραιωμάτων,
[2 > ᾿
ὥστε κατ᾽ αὐτὸν τρισσὸν εἶναι τὸν
ἥλιον,
a 4 3 4 “.
τὸ μὲν ἀπὸ τοῦ αἰθερίου πυρός, τὸ δὲ
3 > + fd ~
dm ἐκείνου πεμπόμενον ἐπὶ τὸν ὑελοειδῆ
t + ? ~
ὑπ᾽ αὐτοῦ λεγόμενον ἥλιον,
5 δὲ 5 4 ~ é 4 ὔ ᾿
ro δὲ ἀπὸ τοῦ τοιούτου ἡλίου πρὸς
ς --
ἡμᾶς πεμπόμενον.
3. Ene Camels af Pritioidns
idea is obviously that of the burning-glass, which became familiar
precisely in the late fifth century; Gorgias speaks of its πόροι, through
which fire penctrates.4 Diogenes of Apollonia secms to have had a
quite similar idea: the sun is a body “like pumice stone,” on which
“rays from the acther concentrate.”*° Here again, the acther is the
source of the sun’s fire, the visible sun only a transmitter and amplifier,
and the passages or “pores’’ of the pumice correspond to the “narrow
interstices” of Philolaus. There are also other points of agreement
between Diogenes and Philolaus.”* But Empedocles as well had a
peculiar theory of the sun, and, in spite of various contradictions in the
evidence, it is clear that it was similar to that of Philolaus.2? Circling
about the earth, he thought, were a fiery and a dark hemisphere. The
fiery one is the source of the sun’s light and heat, whereas the sun we
see is a phenomenon of reflection, or maybe a burning-lens. Though
it is only by conjecture, at best, that a chronological order can be seen
in this complex of solar theories,?® it is at least certain that they all
belong closely together. Obviously the realization that the moon has
no light of its own?® was exerting an influence in impelling people
ra
Aétius and Achilles supplement one another. The “mirror” mentioned by Aétius must
be an error, as διηθεῖν shows; so that the report of Achilles is the more reliable (Boeckh
127; Martin, Philolaus 135f; cf. also Tannery, HScH 237f; Heath, Aristarchus 117). Por
πῦρ περιέχον sce Philolaus A16. Cf. also Guthrie I 2858.
24 Gorg. fr. 5: ἀπιέναι τὸ πῦρ διὰ τῶν πόρων. Cf. Diels, SBBIn 1884, 343f, Theophr.
De igne 73f (the burning-glass and burning-mirror classed together under the heading of
ἀνάκλασις). Ar. Nub. 767-769.
25 Diogenes of Apollonia A13 = Aét. 2.20.10: κισηροειδῆ τὸν ἥλιον, εἰς ὃν ἀπὸ τοῦ
αἰθέρος ἀκτῖνες ἀποστηρίζονται. Cf. At2 and Ar4, on the stars and the moon. Epicurus
fr. 343 Usener = Ach. Is. p. 46.18 Maass: κισηροειδῇ (τὸν ἥλιον)... ἐκ πυρὸν διὰ τρημάτων
τινῶν τὸ φῶς ἐκπέμποντα.-- -ὙΒε αἰθήρ is different from the air, according to Anaxagoras
(A7o, cf. frr. 1, 2, 12); it is warm, and surrounds the earth. Diogenes of Apollonia also
speaks of the θερμότης τοῦ αἰθέρος (Ach. Is. p. 40.9 Maass).
26 Cf. above, ch. IV 1, n. 112; below, n. 36.
27 Frr. 44, 430, A56; cf. Burnet, EGP 238f; G. Kafka, Philologus 78 (1923) ΙΔ (p.
213: the introduction of glass in this period); Kranz, Hermes 73 (1938) 103, and Emped.
50. A56 is very close to A30 in the description of the two hemispheres, but seems to say
that the sun is in the dark half of the sky. Kafka and Kranz mention the burning-glass;
but it is striking that all the testimonia speak of the role of the earth in the ἀνάκλασις
phenomenon. Can Empedocles be thinking of reflection, after all—the sun as a sort of
mirror-image of earth in the sky?
28 The fact that the doctrine of Philolaus is relatively clearest is duc to Achilles’ omission
of the corresponding theories of Empedocles and Diogenes. Philolaus was probably
younger than Empedocles. Burnet (EGP 298) and Heath (Aristarchus gof) believed that
Empedocles’ theory of the sun was earlier, Zeller (I 982, 1027) the reverse.
28 Anaximenes A16 = Eudemus fr. 145 (Tannery, HScH 216ff, can hardly be right in
questioning the authenticity of this; cf. Gigon, Ursprung 108), Anaxagoras A76, Empe-
docles frr. 43, 46, A30; on Parmenides (frr. 15, A42) see Jaeger, RhM 100 (1957) 42ff.—
Ion, DK 36A7, called the moon ὑελοειδές.
343
See LS ee τὸν Sane Ses Ὑππαμοὃο ΞΘ ΦΨΟ Β5 ἊΣ ee Oe VS SS Sewre Se le eve ye
to look for foreign sources of the sunlight and starlight as well.2°
Even if it were not certain that the historical Philolaus lived in the late
fifth century, we should have had to assign the theory of the sun in the
Philolaic system to that period.
The report of Aristotle and Philip of Opus, that the higher frequency
of lunar than of solar eclipses was explained by the presence of the
counter-earth, and perhaps also other earth-like bodies in space, takes
us into a similar context.3! Sometimes the earth, and sometimes the
counter-earth, shuts off the sun’s light from the moon.®? This astrono-
mical use of the counter-earth has often been played off against
Aristotle’s ironical statement that the counter-earth was only invented
to bring up the number of celestial bodies to ten;#* but from an
astronomical point of view, this explanation of lunar eclipses is
unsatisfactory and betrays a lack of exact information. In any case it
is not Pythagorean in origin. Anaxagoras had invisible, dark bodies
circling the earth below the moon, responsible, along with the earth’s
shadow, for eclipses.% Diogenes of Apollonia followed him,?* and,
earlier, even Anaximenes had assumed such “earth-like” bodies.3?
Once more a feature of the Philolaus system leads us into the ambit
of fifth-century φυσιολογία; it is impossible to separate “Ionian” and
“Pythagorean” astronomy.
Even the most exciting idea of the Philolaic system, the movement of
the earth, may not be unexampled. Leucippus declared: τὴν γῆν
9° “Men saw reflected light everywhere,” says Burnet (EGP 239, writing on Empe-
docles); cf. Metrodorus of Chios, DK 70Ag (the fixed stars illuminated by the sun; also
τὴν τῶν ἄστρων ἀνταύγειαν, Hebd. 1.2), and Hippocrates (above, ch. IV 1, nn. 113, 11 5).
5? Solar eclipses are equally numerous, but always visible from only part of the earth’s
surface.
82 Philip of Opus (from his book Περὶ ἐκλείψεως σελήνης; cf. Suda s.v. φιλόσοφος;
Martin, Philolaus 150) and Aristotle (fr. 16 W., not in Rose) = Aét. 2.29.4: ἀντιφράξει
τοτὲ μὲν τῆς γῆς, τοτὲ δὲ τῆς ἀντίχθονος; cf. Arist. Cael. 293b24.
88. Against Aristotle (Met. 98628), Burnet (EGP 305) alleges that the counter-earth
was “a hypothesis intended to account for the phenomena,” and Cherniss (Pres. 199)
says that Aristotle in another passage shows that “‘he is aware of the baseless captiousness
of his former criticism.”
*# Emphasized by Martin, Philolaus 150. It has been conjectured that the “dark bodies”
are introduced to explain the rare phenomenon of the sun being visible during an eclipse
of the moon, on the opposite horizon (Cleomedes 2.6; Heath, Aristarchus 79f; Boll,
RE VI 2351; actually, it is a phenomenon of refraction). But in that case the body casting
the shadow would have to be above the earth, not below it as the Pythagorean counter-
earth is; and the Pythagoreans would have spoilt the special point of their hypothesis.
That they had this very uncommon phenomenon in mind is, however, unlikely.
3° Anaxagoras A42 §§6, 9, A77 = Theophr. Phys. op. fr. 19, Dox. 493.
36 Diogenes of Apollonia A12.
3” Anaximenes A7, 14.
344
a ae Waele UF δ ΘΗ
ὀχεῖσθαι περὶ τὸ μένων δινυυμένη»,ὉΝ Here, too, it must be a question of
unobservable motion. ‘The carth takes part in the vortical whirl,
though more slowly than the bodies dancing about it. Anaxagoras
also had the carth taking part in the cosmic whirl, if, as he says, the
stars are stones thrown off from [1.39 In Democritus’ view, the earth
at first ““wandered about,” then later became dense and heavy, and
settled to rest.4° Like Leucippus, he thought the celestial pole originally
stood at the zenith; then later the earth tilted toward the south, so
that it became the “north pole.” This means, in principle, the ex-
planation of a celestial phenomenon by alteration of the position of
of the earth.
The first, intuitive conception of a theoretical hypothesis isa different
thing from its development into a precise and verifiable theory.
Parmenides can hardly have had more reason than his postulate of
symmetry for supposing the earth to be spherical; and the first thought
of the movement of the earth was also probably not an attempt to
explain specific phenomena. It may have sprung, simply, from the
wish to assign the earth a position as a fully-fledged member of the
cosmic process. Only when people, making a new start, tried to
discover physical laws to explain movement, did those arguments
emerge which, from Aristotle to Ptolemy, seemed to refute the
hypothesis of a moving earth.
The principle of cosmic uniformity lies behind the expression in
Aristotle that the earth is “one of the stars.”4? But this leads im-
mediately into a thicket of mythological connections. The converse of
38 Leucippus Ar §30. Schiaparelli objected to this, that the fixed stars also do move
(§33; Schiaparelli, Vorl. 21 ἢ. 37); but compare the same inconsistency in the system
of Philolaus.—On the other hand, Anaximander A26 = Eudemus fr. 145, ἡ γῆ μετέωρος"
καὶ κινεῖται περὶ TO τοῦ κόσμου μέσον, is probably corrupt (καὶ ἀκίνητος). See DK,
n.; Wehrli 120f. It is a misunderstanding according to Zeller I 303 n. 1; Kahn s54f; but
accepted by Burnet, EGP 66 n. 3; Heidel, CP 1906, 279ff; Taylor, Tim. 164; Mondolfo,
Inf. 311 n. 1.
39 Burnet, EGP 269 n. 2, citing Anaxagoras A7I and fr. 16.
40 Democritus Ag5: κατ᾽ ἀρχὰς μὲν πλάζεσθαι τὴν γῆν...
41 Leucippus A27, Democritus Ag6 (cf. Anaxagoras ΑἹ §9). Kranz (RhM 100 [1957]
122ff, Emped. 50) concluded from fr. 48 that Empedocles believed in rotation of the
earth. Plato (Symp. 190b) says that sun, moon, and earth are all round and in motion
(Kranz, ABG 2.46 n. 29), and at Crat. 397d says that sun, moon, earth, stars, and sky are
“all together” eternally in movement (Boyancé, REG 1941, 146); this might be a matter
of direct Pythagorean influence. Arist. Cael. 293a18, τῶν πλείστων ἐπὶ τοῦ μέσου κεῖσθαι
λεγόντων (τὴν γῆν), ὅσοι τὸν ὅλον οὐρανὸν πεπερασμένον εἶναί φασιν, leaves all possibilities
open for those who believe the heaven to be infinite (Anaxagoras, Leucippus, Demo-
critus?).
42 Cael. 293422; cf. Philolaus A16, 21.
345
MW. ASTRONOMY AND PYTHAGOREANISM
this statement, that the stars area kind of earth, is ascribed by Heraclides
to the Pythagorcans.4* Philolaus taught that the moon “is inhabited
all around, as the earth is in our zone, by creaturcs and plants that are
larger and more beautiful, for living creatures on the moon are
fifteen times as strong, and climinate no excrement. Their day is
proportionately longer.” Herodorus of Heraclea, in the fifth century,
wrote that “women on the moon lay eggs, and their offspring are
fifteen times as large as we are.’””45 Herodorus presupposes the story that
Helen, who was born from an egg, had fallen from the moon:4¢ a
similar story was told of the Nemean Lion.4” As support for his theory
that the moon was an inhabited “earth,” Anaxagoras cited not only the
observation of a fallen meteorite, but the story of the Nemean Lion.48
The Pythagorean acusma that the sun and moon are the “Isles of the
Blest” belongs in this context.9
“8 Heraclides frr. 113 (1134: ταῦτα δὲ τὰ δόγματα ἐν τοῖς ᾿Ορφικοῖς φέρεται may be
an addition of the doxographer, as Tannery thinks, MSc IX 221ff), 114, 11 5 (every star
a world to itself with its own atmosphere).
44 Philolaus Azo, τῶν Πυθαγορείων τινὲς μέν, ὧν ἐστι Φιλόλαος . . . An error of
calculation is involved in the statement that on the moon a day is 15 times as long as a
day on earth. This is easy because the word “day” is ambiguous (12 or 24 hours). During
a lunation the moon rotates once on its own axis, so that its “day” is 15 earth days (i.e.
15 X 24 hours!), and its “night” 15 earth nights. (Martin, Philolaus 145ff, imagines a
kind of axial rotation, too complicated to describe here, in order to eliminate the error;
but the Philolaic system in general does not rest on mathematical subtleties.)
38 Herodorus, FGrHist 31F21 = Ath. 2. 571 (also cf. fr. 4). This has been seen as relevant
to Philolaus since Martin, Philolaus 144; cf. DK I 404 n.; Capelle 3; M. Detienne, RHR
158 (1960) asff. There is no way of knowing whether Herodorus is drawing on Philolaus
or on older tradition.
* Neocles of Croton (otherwise unknown; a Pythagorean?) ap. Ath. 2.57; cf. Jacoby’s
notes; Eustathius p. 1488. 19ff(on Od. 4.121); cf. M. Detienne, “La légende pythagoricienne
d’Héléne,” RHR 152 (1957) 129-152.
“7 Herodorus FGrHist 31F4 = Tatian 27; Epimenides DK 3Bz.
“8 A77. This may be a target of Democritus’ charge of plagiarism (D.L. 9. 34). Philolaus
is thus not the first to speak of the moon as inhabited (pace Gundel, RE XVI 770); cf.
also Orph. frag. 91. Xenophanes A47 remains uncertain (see DK, n. ad loc., and Gigon,
Ursprung 172f£). The stories of children of the moon are of a somewhat different character.
Musaeus was a son of Selene (PI. Rep. 364e, Hermesianax 2.1 5 Diehl, etc.), and a verse
attributed to Epimenides runs, καὶ yap ἐγὼ γένος εἰμὶ Σελήνης ἠυκόμοιο... (DK 382,
attributed to Musaeus by Diels, DK I 33 n.; O. Kern, Religion der Griechen II [Berlin, 1935]
175, conjectures that Epimenides introduced Musaeus speaking; but in any case the words
καὶ γὰρ ἐγὼ... show that there were other members of the moon family. For the
prophet’s reference to h!s own divine origin and experience, cf. Empedocles fer. 11 4.13,
117.1)
*® Above, ch. IV 1, n. 109. This could be connected with the legend of Helen (who
ends up in Elysium) as well as with Musaeus and Epimenides. In each case are found both
the origin from a higher realm and the return to it.—Plato (Symp. 190b) speaks of the
human race as descended from sun, moon, and earth. Cf. Capelle 3; Cumont, Symb.
182ff-—The relation between the moon and the isle of the Hyperborcans, as depicted
by Hecataeus of Abdera, is interesting (FGrHist 264F7 = DK 73B5 = Diod. 2.47.5):
346
4. The Cosmos of Vhilolaus
No one can give us information about the inhabitants of the moon
and the stars except someone who is one of them or who can make
contact with them; and the idea of a “journey through the skies”
brings us back once more into the world of “shamanism.” Certain
details seem to confirm that this is a real connection: the fact that that
other world is “greater and more beautiful” is part of the experience
of ecstasy,®° and freedom from excretory function suggests an existence
apart from the body.*! |
But the counter-earth as well fits into this picture. The same Hero-
dorus asserted that vultures are not native to our earth, εἶναι τοὺς
γῦπας ἀφ᾽ ἑτέρας γῆς ἀδήλου ἡμῖν.53 This cannot mean the moon, which
is anything but invisible. But the Pythagorean counter-carth is a
γῆ ἄλλη, ἡμῖν ἄδηλος,58 so that the ἀντίχθων surprisingly turns up in
fifth-century literature, in a purely mythical context. The vultures of
which Herodorus speaks are rationalization of the griffins—ydmes for
ypomes—which live along the road to the land of the Hyperboreans,
at the entrance to the world beyond, “Gripes Hyperborei, quos =
generat mundus alter.’** Theopompus represents Silenus as telling
of an infinitely great ‘“‘continent outside this world of ours,” whose
inhabitants are twice as large and twice as long-lived as we. All
their laws are exactly opposite to ours. And, when these creatures once
wished to come and visit us, they only got as far as the Hyperboreans
and turned back.** The theme of a “countcr-world”’ where everything
50 Cf. Pl. Phd. 1ogb, Plut. De gen. s90c, De sera 5636. .
51 D.L. 8.19, of Pythagoras: οὐδέποτ᾽ ἐγνώσθη οὔτε Staywpav... Lucian Ver. hist.
1.23, Ctesias FGrHist 688F45 §44 = Phot. Bibl. 48012 (Rohde, Rom. 206 n. 4, 288 n. 2),
Along with this goes the absence of normal nourishment; Lucian’s moon folk subsist
on odors (cf. Megasthenes ap. Strabo 15, p. 711, Plin. HN 7.25, Plut. De fac. 938, scarcely
fortuitous in the context of moon dwellers), as do the dead (Lucian Charon 22, Heraclitus
fr. 98). Perhaps the origin of this idea is in the practice of the shamans, who (in yy)
achieve ecstasy by inhaling certain fumes (Hdt. 1.202, 4.75; Meuli 121 ff, Schol. Luc ian
p- 20.13). Aristotle tells us that, according to the Pythagoreans, τρέφεσθαι... ἔνια ζῷα
ταῖς ὀσμαῖς (Sens. 445416 = DK §8B43); and this could be interpreted as applying to
the inhabitants of the moon. Aristotle himself (Gen. au. 761b21) assumes that there are
inhabitants on the moon (in a similar context to [Pl.] Epiu. 984d), and mentions the
“vultures” of the ‘other earth” (below, n. 52). . .
82 FGrHist 31F22 = Arist. Hist. ant. $63a7. Jacoby, in his note, mentions the moon
but is dubious.
53 Arist. Cael. 293423, b22. . |
δά Apul. Met. 11.24.3. In Lucian Ver. hist. 1.11 the moon men ride on vultures
(ἑἱππόγυποι). Here moon and counter-earth have coalesced (cf. above, ch. Ill 1, n. 78).
δ FGrHist 11§F75 = Acl. VH 3.18. Rohde (Rom. 219fF) sces in this an attempt to
outdo Plato’s Atlantis (καταντικρὺ ἤπειρος, Pl. Tim. 25a); and there is clearly also a
connection with the Phaedo (above, n. 50), in that the “true earth” is larger and more
beautiful. This does not mean, however, that ‘Vheopompus is arbitrarily spinning out
Plato’s statements; rather, he is drawing from the same tradition. ~The “true continent
outside our world, beyond the Ocean, appears also in Plat. f Me fac. YAN b, in the Pythag-
orizing final niyth, as well as Schol, Pl. Ale. raza; cf lambulus in Diod. 2.56.
347
χω πω ee νας EE EE ES Se SE A BE VELA ES
is the Opposite of what we know, is widespread in folklore and has its
cHect on travel literature as well: it is especially common to represent
the realm of the dead in this manner.®? Thus the Pythagorean counter-
carth, which is naturally thought of as inhabited,5® has a real meaning
in the world of myth, and its position in the astronomical system is an
expression of its character. Clearly there is a background for this, in
the tradition of shamanistic narrative, similar to that which lies behind
the story of the inhabited moon. The legend of Pythagoras and the
doctrine of transmigration, as well as the acusma about the moon
and sun as the Isles of the Blest, presuppose just such a shamanistic
outlook, so that we are probably justified in calling the whole
complex Pythagorean—perhaps various features of it actually
originated with Pythagoras himself, and perhaps he was just the
most conspicuous member in a more comprehensive chain of
tradition.
The system of Philolaus, which we found intractable to analysis
as an expression of scientific astronomy, now takes its place beside,
perhaps even before, Herodorus of Heraclea, that is, in the second half
of the fifth century B.c.—precisely the time of the Pythagorean
Philolaus of whom Plato speaks. It emerges from the same kind of
concern as that of Herodorus, the interpreter of myth: ancient lore,
transmitted ἐν μύθου σχήματι, is newly formulated in the terms of
contemporary natural philosophy or φυσιολογία. Thus shamanistic
myths take concrete formas specific components of the world, and the
earth is relegated to equality with them as “one of the stars.” An
“eccentric” attitude toward things, so to speak, a devaluation of
earthly existence in comparison with “ purer” worlds,®® is doubtless
as strong a motive force here as the requirement of symmetry which
makes correlatives of "Ὄλυμπος and Ζανὸς φυλακή, the “Limiting”
which is united in harmony with the “Unlimited.”
°6 dt. 2.35ff: Αἰγύπτιοι. τὰ πολλὰ πάντα ἔμπαλιν τοῖσι ἄλλοισι ἀνθρώποισι ἐστήσαντο
ἤθεά τε καὶ νόμους.
87 L. Lévy-Bruhl, The “Soul” of the Primitive, tr. L. A. Clare (New York: Praeger,
1966) 303f. See also Pi. fr. 129: day for us is night for them, etc.
58 Philolaus A17, τοὺς ἐν exeivn ... G. B. Burch maintains, Osiris 11 (1954) 286-280,
that the counter-earth is necessary to preserve the equilibrium of the universe; but, if
that is so, the inhabited moon would necessitate a counter-moon (288, ἢ. 96).
°° Cf. the acusma τί ἀληθέστατον λέγεται, ὅτι πονηροὶ of ἄνθρωποι, and ἀγαθὸν οἱ
πόνοι... ἐπὶ κολάσει γὰρ ἐλθόντας δεῖ κολασθῆναι (lam. VP 82, 85; above, ch. II 4,
n. 14).
248
3. Ene Cosmos of Patloiaus
Proclusand, after him, Damascius®® report that Philolaus dedicated”
certain geometrical figures to particular gods—the angle of the triangle
to Cronus, Hades, Ares, and Dionysus, the angle of the square to
Rhea, Demeter, and Hestia, and the angle of the dodecagon to Zeus.
Damiascius adds that the semicircle was sacred to the Dioscuri. One
would quickly reject this late testimony, if it were not corroborated
by a very ancient piece of evidence. Eudoxus mentions that in the
Pythagorean doctrine the angle of a triangle belongs to Hades, Dio-
nysus, and Ares, and that of the square to Rhea, Aphrodite, Demeter,
Hestia, and Hera, that of the dodecagon to Zeus, and that of the 56-
angled figure (the hekkaipentekontagonion) to the baneful Typhon.*!
This remarkable doctrine is thus attested for pre-Platonic Pythagoreans
by a contemporary of Plato. Scholars from Boeckh to Zeller scarcely
knew what to make of it,® till Tannery, Newbold, Boll, and Olivieri
pointed out the connection with astrology. According to an. astrological
procedure often repeated, with certain variations, triangles and
squares are inscribed in the zodiac and are then associated with elements
and planets.® There are four τρίγωνα and three τετράγωνα; a triangle
spans four signs, a square three. This seems to explain the striking
connection of three goddesses with the square, and of four gods with
the triangle, in Philolaus (even though this precise correspondence is
not attested in the Eudoxus passage). The dodecagon, which corre-
sponds to Zeus, is the whole zodiac with its twelve signs. Half the
60 Philolaus Arq = Procl. In Eucl. 130.8, 166.25, 173.11, 174.12. Proclus alludes sas
where to the Pythagorean connection of σχήματα and gods (Theol. Pl. 1.4 p. 9; In Parm.
p- 647, 924 Cousin [Paris, 1864]), so that Damascius (Princ. Il 127.7 Ruelle) may have
derived his additional data about the circle and the semicircle from a more detailed
exposition by Proclus which is now lost. In any case the sentence καὶ μήποτε ( perhaps’’)
ws καθόλου εἰπεῖν τὸ μὲν περιφερὲς κοινὸν σχῆμά ἐστιν πάντων τῶν νοερῶν θεῶν ἧ νοεροί,
τὰ δὲ εὐθύγραμμα ἴδια ἑκάστων is Neoplatonic interpretation, probably by Damascius
imself. .
"Ἢ fr. 84 Gisinger = 293 Lasserre = Plut. De Is. et Os. 30.363a. Schaarschmidt
(43ff) maintained, consistently with his general argument, that the oe emul
was spurious. But Boll (Sphaera 472-478) traced other similar indications of astral theology
to Eudoxus. ae
δὲ ἢ 1428, Zeller I 499 n. 1.
63 nie ΕΣ: 2 τ 379-386 = MSc VII 131-139; Newbold 198ff; Boll ἫΝ
fortunately never worked out in detail the hints he gave at N [Jb 1908, 119 (= tis wr
τοί, repeated 382). Olivieri 30ff. The τρίγωνα are associated with the 4 scasons and bes 4
elements, though the goddesses named by Philolaus cannot be related with them direct y,
nor with the zodiacal signs. Olivieri tried to carry the interpretation out in detail. Boll
pointed out the connection of the planet Jupiter with the dodecagon. Newbold hazarded
an astronomical interpretation of the connection between Typhon and the §6-angled
figure (207ff). Cf. the 28-day month (Plut. De Is. ef Os. 367f).
349
iV, ASTRONOMY AND PYTTLACC IREANISM
sighs of the zodiac become visible ina single night, and this suggests
the semicircle, which is sacred to the Dioscuri. These sons of Zeus live
and die on alternate days. The orbital period of Jupiter is almost
exactly twelve years (11.86), so that in a single year it traverses, approx-
imately, one sign. Dodeketerides built on this are ascribed to Zoroaster
and Orpheus; van der Waerden relates them to the “primitive
zodiacal astrology” developed in Babylon in the sixth century B.c,65
Frank insisted that no traces of astrology can be detected in Greece
before the time of Plato’s old age; yet not only Eudoxus®? but
Ctesias as well speak of the Chaldaeans’ divination by the stars,68
and reflections of astrological ideas have been detected in the Hips
pocratic Regimen. If the zodiac was introduced to the Greeks from
Babylon before 500,79 and the planets about 440, one would infer
the presence of the related astrological conceptions even without
specific evidence. In the mélange of myth and φυσιολογία which
Philolaus’ astronomy proves to be, we also find the first traces of
astrology; though here the Babylonian tables are replaced by the
idea of the angle—the graphic, geometrical, that is to say, the specifically
Greek” element.
4. HARMONY OF THE SPHERES AND ASTRAL IMMORTALITY
A very famous and impressive image, uniting the disciplines of
music and astronomy, is that of the “music of the spheres.”’ It is
" pact Plac. Theb. 4.527.—The sky is called Zeus (Empedocles fr. 142.1, Democr.
r. 30; cf. Critias fr. 25 and later Aratus 1ff, Ennius fr. sc. 345 Vahlen, Macrob. Somn.
Se. 1.17.14). He is so as fire (Empedocles fr. 6.2), air (Diogenes of Apollonia A8, Eur.
Tro, 886); Διὸς αἰθή
ee Pere αἰθήρ, Hdt. 7.8y, Eur. fr. 839: also see Hebd. 6, above, ch. IV 1, n. 66.
86 Frank 281f.
47 [ἐμ secerre τς .- ἢ; «ς Coq
7 : . 343 Lasserre = Cic. Div. 2.87: “Chaldaeis in praedictione et in notatione culusque
Vitae ex natali dic minime esse credendum.” Reports of dubious authenticity are given
by ps.-Arist. fr. 32 = DL. 2.45 and Gell. NA 15.20.2. :
"ΟΠ list O88F1 §§23~2 5.
"W. Capelle, “Alteste Spuren der A ie bei I
; strologie bei den Griechen.” Herne
ἜΡΩΣ 5 cnen, Hermes 60 (1925)
ΤΟ Above, ch. IV 2, nn 53-55 R. Boker, RE XX
, nn. Ὄ , RE ΠῚ 876, on the ground of a dubj
conjecture in Plin. HN 2.31, ascribes to Cleostratus the zodiacal a μων
4. Harmony of the Spheres and Astral lnmortality
attested by Aristotle! and indirectly by Plato,® as being a doctrine
of the Pythagoreans, who interpreted the Sirens of mythology as the
makers of this cosmic music. Scholars have long thought that the
music of the spheres had no organic connection with the Philolaic
system.* It is difficult to relate the ten revolving celestial bodies to
music.’ The source of the whole idea, it is thought, is rather the associa-
tion of the ancient and proverbial “seven-stringed” lyre® with the
later, but no less well known, idea that the planets are seven in number.”
The general nature of the Pythagorean musical theory is rightly
cited: the coherence of number and sound. Harmonic intervals
* Cael. 290b12ff (“the Pythagoreans” named at 29148), fr. 203. Archytas fr. 1, Plut.
Mus. 1147a. The expression “harmony of the spheres” is inappropriate, strictly speaking,
as applied to the time before Eudoxus, for then one spoke of bodies, wheels, rings, circles
in the sky, but not yet of spheres (Burnet, EGP 110).
* Rep. 530d, Οὐαί. 405c; also Rep. 617b, in the myth of Er. On the Timaeus, see below,
n. 19.
3. On the acusma of the tetractys, see above, ch. II 4, ἢ. 154. Pl. Rep. 617b. On the Siren
idea, see E. Buschor, Die Musen des Jenseits (Munich, 1944); on the Pythagorean inter-
pretation, Delatte, Litt. 132f, 260f. In the Temple of Bel at Palmyra (early Empire) the
Sirens are depicted next to the planetary gods; this is interpreted by L. Curtius, RM 50
(1935) 348-353, as having to do with the music of the spheres. On the other hand, later
Pythagorean tradition spoke of the Muses in relation to the harmony of the spheres and
thought of the Sirens as representing sensuous, worldly music; see Clem. Al. Strom.
1.48.6, Por. VP 39, Demophilus 23 (Mullach I 486), and P. Courcelle, REA 46 (1944)
731 (vs. Cumont, Symb. 329f).—“Pythagoras” himself says, Schol. Od. 1.371 -
p. 172.6 Thesleff, ἔξω γενόμενος τοῦ σώματος ἀκήκοα ἐμμελοῦς ἁρμονίας; and Nicomachus
also records that Pythagoras was able to hear the harmony of the spheres (Por. VP 30 --
lam. VP 66f, with a mistaken interpretation of Empedocles fr. 129).
* Martin, Pyth. 110ff; Tannery, Astr. 327; Zeller I $40 n. 2; van der Waerden, Astr.
29. Tannery (cf. MSc. VII 158f) puts the harmony of the spheres later than Philolaus,
though usually the Philolaic system is thought of as a later development. It is not important
that the surviving fragments of Philolaus donot refer unambiguously tothe cosmic music;
the doxographers would hardly have had occasion to include this under the rubrics
περὶ τάξεως ἀστέρων (A16), περὶ ἡλίου (A19), or περὶ γῆς (A21). We do find the expression
χορεύειν in Ατό; and Aristid. Quint. p. 145 M. = 119.27 W.-I. finds in the words ἄστρων
χορός an allusion to τὴν τῶν πλανήτων ἐμμελῇ κίνησιν.
δ Though before the time of Timotheus, Histiacus of Colophon had introduced a
ro-string lyre (Nicom. Exc. 274.4 Jan).
δ Sarcophagus of Haghia Triada, Hymn. Hom. Merc. 51, (Terpander) fr. 4.4, Pi. Pyth.
2.70, Nem. 5.24, Bacchyl. 20b2, Ion of Chios fr. 6.3 Diehl, Eur. Alc. 446, lon 881, ete.
See Deubner, AM $4 (1929) 194-200. It is quite uncertain how these 7 strings were tuned;
below, ch. V 2, nn. 34ff.
“Varro Atacinus fr. 14 Morel (Gramm. Lat. VI 60), Philo Op. 126, Nicom. Ench. 4
(cf. Th. ar. 71.15ff), Lucian Astrol. 10, “Orpheus” Orph. frag. T58a .. Serv. Act. 6.645,
Dio Cass. 37.18, Lydus Mens. 2.3. There do not seem to be any older references than these
on the number 7 as a link between music and astronomy. Both are omitted from On
Sevens, as Roscher emphasizes. The 7 analogy as point of departure for the whole theory
of cosmic music is accepted, among others, by Jan, Philologus 1893, 15f; Wilamowitz,
“Die Harmonie der Sphiren,” Reden aus der Kriegszeit U6 (Berlin, 1915) 15; and Gundel,
RE XX 2056.
Γι EEO ΣΝ ΥΩ E FINES ΓΚΥΓΙΠΑ( ἸΚΕΑΝΙΝΜ
correspond τὸ harmonic relationships of distance and velocity;# and
since a musical tone—as distinguished from a mere noisc—implies a
uniform motion, one can infer from all this a Pythagorcan system of
astronomy, in which the plancts—all seven of them, long known—
circle about the earth in uniform movenients, at various distances from
one another. This Pythagorean astronomy, to be seen in Plato, is
thought to go back earlier than Philolaus.® If his system belongs to the
fifth century, then “ Pythagorean astronomy’ had reached unexampled
heights even earlier; and, if the latter cannot be dated earlier than
Archytas, then “Philolaus” becomes a forgery—one produced,
however, before the time of Aristotle.
It is striking that Aristotle and Alexander of Aphrodisias, the latter
citing Aristotle’s now lost book on the Pythagoreans, make no
distinction between the system of Philolaus and the assumed, very
different Pythagorean astronomy.° When one considers also that
before Eudoxus there was no mathematical theory of the planets
based on the concept of uniform circular movement, one cannot
help suspecting that it is a mistake to assume the existence of an earlier
mathematical astronomy that served as basis for the idea of the music
of the spheres. Actually, the view, put forward again and again as
virtually self-evident, that the thought of cosmic music was an in-
ference from some kind of scientific knowledge," is a misconception.
None of the explanations of the celestial gamut known in the
tradition of later antiquity has any claim to be authentic. Plato speaks
in the Republic of “a harmonia” of eight tones, and in his Hermes
Eratosthenes presented this as a scale one octave long” Thus the
oldest sources speak not of seven but of eight notes, as the Pythagoreans
too, in Aristotle’s account, include the sphere of the fixed stars in the
cosmic orchestra.!3 The scale of the music of the spheres which in
later times was most widely known and was attributed to Pythagoras
himself is actually a botched version of Eratosthenes’ exposition,
8 Arist. Cael. 290b21, fr. 203.
"ΟΝ Frank 30ff; van der Waerden, Astr. 26, 29ff; Junge, C&M 1947-1948, 18 3ff.
19 Alex. Met. 41.2ff = Arist. fr. 203 expressly relates the harmony of the spheres to the
“Io moving bodies.”
“| According to Tannery, Astr. 328, the harmony of the spheres was deduced from (or
“followed,” découlée) the idea of the kinship of the sciences; according to Junge, CGM
1947-1948, 185ff, from the discovery that the planets have fixed orbital periods.
12 Pl. Rep. 617b, Eratosthenes fr. 15 Powell = Theo Sm. 105.15, Anat. p. 38 (Th. ar.
75.6); cf. Theo Sm. 142.7ff = Chalcid. 73.
18 Arist. Cael. 290b18: τοσούτων τὸ πλῆθος ἄστρων... φερομένων, cited by Zeller I
$41.2; Heath, Aristarchus 108.
352
4. Harmony of the Spheres and Astral Immortality
created by the desire to build cosmic measurements inter 11... An
attempt in Cicero's Somminm Scipionis'® to combine eight spheres with
seven notes is a patchwork compromise. Nicomachus is at least more
consistent, in climinating the movement of the fixed stars and basing
the system on the individual movements of the plancts in the zodiac,
so that the moon being fastest, is credited with the highest note;
but in both respects he contradicts the reports of Aristotle.1® The most
lucid and consistent system of cosmic harmony, employing only φθόγγοι
ἑστῶτες, makes no claim to be derived from the ancient τγδάϊοῃ. 7
Alexander intimates that Aristotle himself knew no detailed ex-
position of the Pythagorean cosmic harmony. The distances of the
heavenly bodies from one another (“intervals”), Alexander explains,
have a certain ἀναλογία; their velocities correspond to these distances,
and the tones to the velocities. Thus the tones themselves have a
“harmonic” relationship, determined by number: “Thus, the
14 ilologus 1961, 31ff. .
16 ΕΣ Seer el ea cae seem to have the same tone (“in quibus serine
vis est duorum’”’), as also in Ptolemy (below, n. 17); cf. Macrob. Somn. Sc. 2.4.9 Sai a
is wrong, Mus. 1.27). Boyancé, Songe 111f, interprets this expression as ΠΔΕΆΠΙΗΡ t at oe
moon and the fixed stars make an octave (cf. Por. In Prol. 104.7, on the octave: ee
ἐστιν ἡ αὐτή; similarly, Arist. Pr. 19.14, 19.18); contradicted by O. es PhW 5 ve)
491ff. There remains in any case the question how the sun can have a different te r
its two ἐσόδρομοι. At Plut. De an dade τυρόν, Venus, and Mercury are obviously
o that there remain only πέντε διαστήματα. -
sere Sat 3 p. 241 Jan, and following him Boeth. Mus. 1.20, 27. This " ite .
be the original view by Jan, Philologus 1893, 17f, and van der Waerden, Astr. " ; re :
too (31), thinks the moon had the fastest movement. An apparent ee or se
that here Hypate, taken literally, is the “highest” note (Saturn); but t ¢ names 0
notes are derived from the way the instrument was held. In defense of his ages : τὸ
the ancient Pythagoreans derived the harmony of the spheres from the in ay en
movement of the planets in the zodiac, van der Waerden (Astr. 36f) cites Arist. r. ae
but he quotes only a part of it and overlooks the fact that, later on, more rapid na
is attributed to the more distant planets. Against Nicomachus statement t τ the
synemmenon system was original, see below, ch. V 2. The Excerpta ex Nicomacho cmp aie
that, differently from Nicomachus, “the first” (of δὲ δὴ πρῶτοι) made Saturn the Nete
te . 272.9 Jan).
εν eerie ἐὺ Plut. i dane (ibid. 1029a-b, a somewhat different —
based on the ς tetrachords); Ptolemy, Canopus inscription (Op. min. ed. oe .
154f; authenticity contested by Jan, Philologus 1893, 35ff; Hoeg, Gnomon 6 παν a
defended by Diiring 1934, 280ff; cf. van der Waerden, RE XXIII 1818-1823; the ce
᾿ sponding exposition in the Harmonics of Ptolemy is lost); Th. ar. 75.8ff = Exe. Neap. 2,
p. 412 Jan (Mus. scr. gr.), with the superscription Πτολεμαίου μουσικά; feline
§84. The notes are, transcribed, A Be abd’ εἰ δ' b’, corresponding tot ie ein ἡ
16 18 21} 24 32 36. Since 214 (found in Ptolemy, Canopus inscr.) is not a whole erin
most have simply substituted 21, which is musically wrong. The eanepy ἀπε» εὐ
distinguishes earth (8 = A) from air (9 = B), and in compensation gives cat εἰ
Mercury only a single note (16 = a); the others count 8 celestial pee ᾿ 3
system is older than Plutarch, it could go back to the mathematician Hypa es, who,
according to Ach. Js. τό p. 43.9 Maass, wrote on the harmony of the spheres.
353
ἐγ, ASTRONOMY AND PYTHAGOREANISM
distance of the sun from the earth being, say, double the distance of
the moon... they considered that there was some arithmetical ratio
in the case of the other planets as well...’ Alexander introduces
his exposition with φέρε εἰπεῖν, that is, “supposing,” or “say, for
example.” The figures given belong to a hypothetical case, cited for
clarity’s sake, and do not belong to a traditional account. If Aristotle
had credited the Pythagoreans with an unambiguously described
scale, Alexander would not have used a fictitious example.
Nor does the structure of the world soul in Plato’s Timaeus signify
a system of cosmic harmony, in which each planet had a note corre-
sponding to its “distance.’’ The seven planetary orbits are derived by
sixfold division of the inner sphere of the heavens, which is dominated
by the nature of the “other.” This division is made “according to”
three powers each of the numbers 2 and 3, which had previously been
developed; but the game he plays with 6 and 7 seems to suggest that
one ought not simply to identify the 7 numbers with the 7 planets
and 7 notes.!® To be sure, musical theory is involved,?° but Plato says
expressly that the movement of the world soul takes place ἄνευ φθόγγου
καὶ ἠχῆς (370); the mythic image of the Sirens, from the Republic, is
translated into the realm of the immaterial and abstract. At the same
time the naive thought of the Pythagoreans reported by Aristotle,24
15 Alex. Met. 40.3ff = Arist. fr. 203, tr. Heath. Heath saw the significance of φέρε
εἰπεῖν (Aristarchus 111 ἢ. 2). Tannery, too (Astr. 327) believed that all detailed expositions
of the celestial music were late.
1 Tim. 36d: σχίσας ἐξαχῇ ἑπτὰ κύκλους ἀνίσους κατὰ τὴν τοῦ διπλασίου καὶ
τριπλασίου διάστασιν ἑκάστην, οὐσῶν ἑκατέρων τριῶν, depending on 35b-c. On the
contradictory ancient attempts to relate the 7 numbers of the Timaeus with magnitudes,
velocities, and intervals of the planets, cf. Plut. De an. proct. 31.1028a, Chalc. 96, Macrob.
Somn. Sc. 2.3.14f, Procl. In Tim. I 212, Taylor, Tim. 161ff. Like Zeller (II 1. 779 n.),
Boeckh (K/Schr Ill 168), and Duhem (53), Taylor comes to the conclusion that *Timaeus
is giving us an estimate of the radii of the planetary orbits in terms of the diameter of the
moon’s orbit” (163). Heath (Aristarchus 163) and Cornford (Tim. 79) are rightly dubious
of this. Like all musical relationships (cf. 36a-b), all numerical relationships are ultimately
derivable from the numbers given; the basic rules of the “system of derivation” (above
ch. I 1) are presupposed. Junge (CGM 1947-1948, 183ff ) attempts an interpretation that
makes sense astronomically: we have “an expression of the orbital periods of all the
planets by means of the numbers 2 and 3 and their multiples.” For example, the moon
takes 27 days, the sun 729 (= 275) half-days, Saturn 27 years, Mars 2, and Jupiter 12 years.
It is likely that Plato’s thoughts were running in this direction(Junge refers to Rep. 588a);
but the basic idea is more important than any specific result—the problem is more import-
ant than the solution.
“0 This was denied by Rivaud, Rev. hist. philos. 3 (1929) 16ff (so that he found it necessary
to brand not only the Philolaus fragments but those of Archytas post-Platonic forgeries:
he seems to overlook, on pp. 6f, that Plato himself, at Rep. 530, expressly mentions
Pythagorean musical theory). The harmony of the spheres, he thinks, is merely crude
interpretation of the Timaeus. (Similarly Moreau, Ame 55.)
21 Cael. 290b15ff.
354
4. Ularmony of the Spheres and Astral limortality
that the tremendous bodies of the stars could not move without making
a noise, is rejected, In the Republic Plato demanded, in explicit polemic
against the Pythagorceans, that the harmony of numbers be regarded
in itself, and without reference to audible sounds, and he carries out
this program himself in the Timaeus.22 Thus the relationship between
astronomy and music is traced to its ultimate, immaterial principle.
The numbers in their order are the basic principle of the cosmos,
which is at the same time “beauty” and reflection of “the Good.”
Therefore the basic constituents of matter, the elements, are also
derived from the process of geometrical arrangement. All this, as
Aristotle and Plato make clear, is not Pythagorean but Platonic.
The contradictory nature of the later tradition, too, sometimes using
seven,” sometimes eight,*4 and occasionally three musical notes,?° and
even connecting the nine Muses with the sphcre of the All,?6 shows that
the idea of cosmic music is not bound to any particular astronomical
system. This concept has nothing to do with mathematical or musical
theory, but comes from a deeper root; and this is why it was able to
outlive even the Ptolemaic cosmology. Of course this carries the
implication that any conclusion about certain astronomical theories
of the Pythagoreans, on the basis of the harmony of the spheres, is
hazardous in principle; we need not suppose that any detailed system
formed its basis.
There are also other conceptions of cosmic music, quite different
though related. One need not speak of planets, or of a many-storied
universe. No matter how things are divided, the correctly attuned car
will hear music.?” The ancient lyre had only four strings, “in imitation
22 Rep. §31c; cf. below, ch. V I. . . ᾿
= Aba n. 7. A connection was also seen with the 7 vowels: Nicom. Exe. 6 p. stl
Jan; Schol. Dion. Thr. p. 197.33ff Hilgart; the Gnostic Marcus ap. Hippol. Ref. 6.48;
Nicor: Th. ar. 71.13ff; Lydus Mens. 2.3; CIG IL 1 no. 2895; cf. F. Dornseiff, Das Alphabet
in Mystik und Magie (Leipzig, 19257) 82f. | . |
24 sia and Eratosthenes, above, n. 12. The arithmological sources (Theo Sm., Anato-
lius) associate the music of the spheres with the passage of Eratosthenes in their discussion
of the number 8, not 7.
25 In the λόγος of the Delphians, Plut. Quaest. conv. 7458. Burnet (EGP 110) and sri
(Philologus 1938, 437) thought that Pythagoras associated the 3 “rings” of oe εἴ
(stars, moon, and sun) with the basic concords of the fourth, the fifth, anc τ Te petayes
26 Τῆς eighth Muse, Urania, fits well; and the ninth was cither ΤΟΙ ΠΠΘΙΗ ἀρ τ νας
Calliope ἃς προφερεστάτη ἁπασέων (Macrob. Somn. Sc. 2.3.1f)—or Aa ey oe
region of carth” (Plut. De an. Procr. 42.1029d, Quaest. conv. 746a). Others looked to the
ποῦ or help (Por. VP 31. | |
ἐ Ὁ. eee He the cee of astrological configurations with μὰν oe
musical concords (Phut. De au. Proce. 31.1028d-c, Ptol. Harm. 3.9). In Haris. 3, A eh
presents a goodly sample of such ingenious fooleries. Cf also Phid. Mus. p. too Kemke.
ne MOTRONOMY AND PYTHAGOREANISM
of the cosmic music, which consists of four elements.”24 The four
scasons correspond in turn to the elements, and stand in concord with
one another: spring makes a fourth along with fall, a fifth with winter,
an octave with summer, “as they say Pythagoras taught.’ All the
sources for this are late, and the part about the clements is consciously
derived from the Timacus.2 But, if Scythinus and Cleanthes called the
sun the “plectrum”’ of the cosmos, this fits better with the thythm of
the scasons, which does depend on the course of the sun, than with
the planctary scale.3! A ἁρμονία of the seasons is spoken of by Euripides,
and then also by Plato.3? Such ideas are traced to oriental beginnings.®3
The idea of a cosmic music, and specifically one connected with the
changing seasons, is widespread; for example, the five notes of Chinese
music correspond to five elements and to the seasons.34 Here one should
not think of direct influence, one way or the other, but of a parallel
development due to similar psychological impulsions; and for this
very treason one seems here to be closer to the root of the idea of
cosmic music than in the context of the planetary scale.
*8 Boeth. Mus. 1.20 p. 206 Friedlein, from Nicomachus. Cf. Clem. Al. Protr. 1.5.1:
fire as νήτη.
29. Aristid. Quint. 3, p- 144f M.: summer is 4, autumn 6, winter 12, spring 8; also
Plut. De an. procr. 31.1028f (“Chaldeans”); somewhat differently Hymn. Orph. 34.10ff
(winter is Hypate, summer Nete, spring the Doric tone: Mese?); a hint only, Hymn.
Orph. 8.9. O. Neugebauer, AJP 63 (1942) 455-488, mistakenly tries to derive a precise
astronomical sense from the passage of Plutarch and to present the actual differences of
the astronomical seasons as harmonic numbers. He overlooks the explanation in Aristid.
Quint. and in addition has to assume a confusion of autumn and summer. Also, all this
speculation obviously takes its departure from the contrast of summer and winter.
* Aristid. Quint. 3, pp. 144f M. Plato himself speaks of ἀναλογία as the δεσμός of the
elements (Tim. 31b-c), and derives from that the fact that they are 4 in number; thus he
knows of a sort of “harmony of the elements”: but the polyhedra cannot be brought
into direct connection with this. What Plato cares about is not a definite and accurate
account, but intimations as to the basic structure of reality.
3! Above, ch. IV 1, n. 107.
δ5 Eur. fr. 943 (thought spurious by Lobeck), Pl. Symp. 188a, Phib. 26a, where Plato
scems to be taking up Pythagorean themes.
δ Plut. De an. procr. 31 speaks of “Chaldeans.” Diodorus reports, of Hermes-Thoth:
λύραν τε veupivny ποιῆσαι τρίχορδον μιμησάμενον τὰς κατ᾽ ἐνιαυτὸν ὧρας" τρεῖς γὰρ αὐτὸν
ὑποστήσασθαι φθόγγους, ὀξὺν βαρὺν καὶ μέσον, ὀξὺν μὲν ἀπὸ τοῦ θέρους, βαρὺν δὲ ἀπὸ τοῦ
χειμῶνος, μέσον δὲ ἀπὸ τοῦ ἔαρος (1.16.1; Hecataeus of Abdera, FGrHist 264F25 [Ὁ];
cf W. Spoerri, Spathellenistische Berichte tiber Welt, Kultur und Gétter [Basel, 1959] 164ff).
When Eratosthenes, in his Hermes, represented this god as inventing the harmony of the
spheres, he was obviously combining Egyptian elements with Greek-Pythagorean material.
G. A. Keller (Eratosthenes und die alexandrinische Sterndichtung (Diss. Ziirich, 1946] 95ff,
y8tf) emphatically denies any Egyptian influence on Eratosthenes. But the 3-stringed lyre
is Egyptian (Keller, 99). Hermes as inventor of the lyre is Greek, but Thoth as inventor
of astronomy is Egyptian (PI. Phdr. 2740).
4@Sce Ε, Μ. von Hornbostel, “‘Tonart und Ethos,” Festschr. ]. Wolf (Berlin, 1929),
I cannot verify Hornbostel’s suggestion of similar ideas among the Arabs. On the Chinese,
sce M. Granet, La pensée chinoise (Paris, 1934) 200ff.
356
4. Flarmony of the Spheres and Astral Immortality
Before the beginning of reflective thought, man feels, in’ various
contexts, an involvement. He unconsciously arranges the multiplicity
of phenomena into a restricted number of schemata. It is the business
of reflection, when it begins, to raise these transitory insights into the
realm of consciousness, to name them, and to assimilate them to one
another. This is how the world becomes comprehensible. In myth and
ritual man tries to make these realizations present and clear, to assure
himself that, in spite of all confusion and all the immediate threats of
his environment, everything is “in order.” It is in such a pre-scientific
conception of order that the idea of cosmic music has its roots; and
number speculation springs from the same soil.
But relationships that usually have their effect unconsciously, or
only enter consciousness as the result of slow and patient reflection,
become immediate, overwhelming experience in ecstasy. The soul that
in ecstasy, or dream, or trance, travels to heaven, hears there the music
of the universe, and its mysterious structure immediately becomes
clear to him. The incomparable and supernatural sound is part of the
same thing as the incomparable beauty and colorfulness of other
worlds.% If Pythagoras was something like a shaman, who in cestasy
made contact with worlds “beyond,” then the tradition that he
personally heard the heavenly music surely preserves something of
truth.°* When we look beyond the facade of analysis and explication
of the harmony of the spheres, what we find is ncither empirical
nor mathematical science, but eschatology. In the religion of Zara-
thustra, the paradise to which the soul ascends is called garo demana,
" House of Psalms.” It was related of Pythagoras that in his dying hour
he asked that the monochord be played: ‘‘Souls cannot ascend without
music, "8?
Attention has often been called to the fact that changes were forced
upon Greek beliefs about the fate of the dead, not least by the develop-
ing understanding of astronomy.*§ If the earth is a sphere, inhabited
35 Ch. ,n. $0.
36 nae ὦ ; Ce also Pl. Rep. 617b, Nechepso-Petosiris fr. 1 Riese = Vett. Val. 6
prooem. p. 241 Kroll: ἔδοξε δέ μοι πάννυχον πρὸς ἀέρα... καί poi τις ἐξήχησεν οὐρανοῦ
βοή; Cic. Rep. 6.18, Plut. De gen. s90c, De fac. 944ab, Corp. Herm. 1.25. .
* Aristid. Quint. 3, p. 116 M. = p. 97 W.-I.; Varro on the lyre of Orpheus in the
Vergil scholium edited by J. J. Savage, TAPA 56 (1925) 235; cf. A. ID. Nock, CR 41
-I7I, 1929) 60. .
see τὰ τ εἰ Cumont (After-life o1ff, Or. rel. 1146), 270f, Symb. passim,
Lux esp. 142ff). Also see Capelle; Pfeiffer, Sterngl. 113ff; Carcopino, Bas. 266fF; Nilsson
It 470ff, Op. 1Π| 250-265; van der Waerden, Anf. 204-252.
357
ἐγ, ASTRONOMY AND ΡΥΤΙΠΛΟΟΚΙΑΝΙΟΜ
all around and limited in size, there is no longer any place for the
subterranean Hades of Homer or for the “Isles of the Blest” in the
distant Occident. In general, the idea hnally prevailed that the Beyond
is in the realm of the stars, that man’s soul caine from the skies and
will return there some day, With many variations in detail, these con-
cepts are dominant from the late Hellenistic period on, both in works
of literature and in the art and the inscriptions found on gravestones.39
Looking for the decisive turning point in this development, modern
scholarship, following ancient tradition, has seen oriental influences at
work—Babylonian and [ranian—and, notably, the influence of
Pythagoreanism.*°
To be sure, Rougier’s attempt to establish the belief in an immortality
among the stars as a deduction from an astronomical discovery of
Pythagoras, making the religious idea a logical conclusion from scien-
tific knowledge, is not convincing in the light of the history of either
religion or astronomy. It is a dangerous oversimplification to
% Cf. esp. Οἷς. Somn. Sc., the lost Consolatio, and Hort. fr. 97 Miiller: Vergil Aen. 6
(with Norden, Vergil VI, 23ff); Plut. De sera 563ff, De gen. §89ff, De fac. 942ff. Tomb
inscriptions are collected by Capelle 33ff; Rougier 108 n. I; R. Lattimore, Themes in
Greek and Latin Epitaphs (1942) 31-43; P. Lambrechts, Hommages Déonna (Brussels,
19$7) 322ff. None of these is complete. On grave symbolism, Cumont, Symb. passim;
On the representation of ascent and descent of the soul in Mithraism, Cumont, Or. rel.
1148} 145f, 290 n. 69.—On the likely role of Posidonius in this connection, see Reinhardt,
Kosmos 276ff, 308ff, RE XXII 778-791.
*° Cumont sees the origin in a fusion of Babylonian astral religion and Persian dualism
(After-Life 95, Or. rel. 272 n. ΟἹ, Symb. 264, Lux 143ff). See also Pfeiffer, Sterngl. 113ff.
Bidez, Eos off, 98f, believes that the origin was Babylonian but that it was transmitted
by the Pythagoreans, and van der Waerden’s solution is similar. With Rougier, Boyancé
(REG 1952, 314f) warns against overhasty derivation from the Orient, and emphasizes
the role of the Pythagoreans, even of Pythagoras himself (349), but also that of more
ancient beliefs and later of Platonism. Nilsson takes a similar line, Op. HI 25off.
41 According to Rougier, the origin of the idea is not the religious imagination of the
easterners, but “Ja révolution astronomique de Pythagore” (2180), i.e., the discovery of
mathematical physics. Pythagoras’ discovery of the contrary movement of the sun
through the zodiac meant that the movement of one of the so-called planets had been
seen as perfectly circular, and the mathematical arrangement of celestial phenomena
recognized. As “conséquences religieuses” (42ff) of this came the doctrines of the presence
of souls in the stars, and of their divinity, of the relationship between the stars and the
soul, because of their eternal movement, of the dualism of heavenly order and terrestrial
confusion; and the reflections of these doctrines in Alcmaeon prove, Rougier thinks,
that their originator was Pythagoras. To this it may be replied, first, that the sun is not a
planet, for the unsophisticated observer, but the paradigm of cosmic order (above, ch.
IV 2, n. 18). Its movement with respect to the zodiac was already known to Cleostratus,
to say nothing of the Babylonians, and the concept of independent movement of the
planets, is, in its basis, older than the Ionian vortex theory (above, ch. IV 2). In all this,
then, there was nothing to discover that might have meant an “astronomical revolution.”
Rougier stopped short, rightly, of ascribing to the Pythagoreans a mathematical theory
of the actual movements of the planets. He recognizes the originality of Eudoxus (26ff),
and does not inject the epicycle theory into the discussion, though he does put forward
358
4. Harmony of the Spheres and Astral Immortality
represent “the Homeric religion” as followed by a “Pythagorean
revolution.” In people's beliefs about the afterlife, there are present
from the very beginning a large number of overlapping and contra-
dictory themes; even in Homer, the idea of the musty ‘House of
Hades,” hateful to the gods, coexists with that of Elysium and with the
apotheosis of Heracles.** The mystery cults bring man the hope of
escaping death and joining the gods; and it is an easy step from this to
the doctrine that man is of divine descent and returns at death to his
place of origin.“4 The association of gods and sky is primeval and
seems self-evident. Artistic representations from the archaic period
show the journey of the deified dead into the Beyond. A team of
winged horses may provide escort “το Heaven,’’4* whereas sea creatures
—Tritons, Nereids, dolphins—point to the Isles of the Blest, beyond
the vague suggestion that the Pythagoreans may have tried the same kind of explanation
for the other planets as for the sun (22); but here he fails to notice that the passage of
Theo he cites (1§0.12ff) anachronistically introduces the epicycles. Rougier overlooked
an important piece of evidence, the acusma about the planets as ᾿ hounds of Persephone,
which does provide a connection between Pythagorean teaching and the oriental ideas
alleged to be so radically different (above, ch. IV 1, n. 109). For argument against Rougicr,
sce also Cumont, Symb. 116 n. 3, Lux 147 ἢ. 1.—Rougier republished his 1933 monograph
almost unchanged in 1959 (La religion astrale des Pythagoriciens), but omitted the references,
so that the older work has been cited here. .
42 A Maori can depict the dead as descending into the underworld, while at the same
time maintaining that they live in the tenth heaven, says L. Lévy-Bruhl, The “Soul
of the Primitive 175. Thus in the Trophonius ritual Timarchus travels into the depths and
is by virtue of this translated to the stars (Plut. De get. 22.590b; cf. Vergil Aen. 6, and
below, n. 46.) . .
43 The Elysium idea itself shows the blending of lightning worship and the myth of
the Isles of the Blest (Burkert, Glotta 39 [1961] 208ff). The apotheosis of Heracles in
doubtless very old, though the verses about it at Od. 11.601ff and Hes. fr. 24.26.1}
M.-W. were athetized in antiquity. |
44 For τὸν λοιπὸν χρόνον μετὰ θεῶν διάγειν as a goal in the mysteries, see PI. Phd.
81a. Also the Gold Plates: θεός ἐγένου ἐξ ἀνθρώπου (DK 1818, 20) and καὶ γὰρ ἐγὼν
ὑμῶν γένος ὄλβιον εὔχομαι εἶναι (DK 1B 18.3, 19.3; cf. 17-7, 17.6, 1743.)
45 βεοὶ οὐρανίωνες, cf. the relationship of root and idea in Zeus, deus, dies, On Olympus,
above, ch. HI 2, n. 31. Dionysus, too, is brought into connection with the stars: ἰὼ πῦρ
πνεόντων χόραγ᾽ ἄστρων, Soph. Ant. 1147; cf. Diod. 1.11: Εὔμολπος μὲν ἐν τοῖς βακχικοῖς
ἔπεσί φησιν" ἀστροφαῆ Διόνυσον ἐν ἀκτίνεσσι πυρωπόν. Musacus, the father of Eumolpus,
is a son of Selene (above, ch. IV 3, n. 48).
46 Etruscan gems (Furtwangler, Gemmen I pl. 7, nos. 1, 2, 3, ca. 600 B.C.) were com
pared with the Phaedrus myth by Delatte, Litt. 74f; the iconographic motif of the team
of winged animals was, however, taken from the Orient, and originally does not have
sepulchral connotations; sce the examples published by J. Boardman, Antike Kunst 7
(1967) pl. 2; but it acquired these by the 6th century B.C.; sec, on the bronze chariot 0
Monteleone in New York, R. Hampe and E. Simon, Griechische Sagen in der frithen
etruskischen Kunst (Mainz, 1964) 53-67; generally, Mculli, Bachofen VII 4o8f; Εἰ Matz,
Gnomon 33 (1961) 63-65. On the throne of Amyclac, Hyacinth and Polyboea were
represented “going to heaven” (ttme of Croesus: Paus. 3.19.4), although at the same
time Hyacinth receives his death offerings through a door in the altar (Paus. 3.19.3).
Cf. also Wuilleumicr 357, 4477
359
IV. ASTRONOMY AND PYTHAGOREANISM
the Ocean. The journey to the Beyond is outside of all geography. In
any case, the notion of divine origin and of return to heaven was
widespread, at least in germ, before the time of Pythagoras. Another
clement present is the divinity of the stars, which inevitably gained in
significance through contact with Babylon.* Then there is the popular
belief in a connection between man and star,4® which had taken on a
special form in Egypt,*® and the general belief in ghosts—that is, in the
spirits of the dead and their proclivity to roam about through the
atmosphere, especially at night, so that in a mysterious way they have
a special relationship to the moon. And, finally, there is the Iranian
conception of the soul’s making a journey to heaven.3! The position of
Pythagoras and the Pythagoreans in this complex tangle is a question
that cannot be settled by general considerations, but only by analysis
of the evidence.
The idea of astral immortality is attested a good number of times
for the fifth century sic. Aristophanes alludes to the doctrine that
people become stars after death;52 Alcmacon put the eternal motion
* Cf. Wilamowitz, GldH I 2$3ff = I 248ff, 2nd ed.; Nilsson, Op. III 31-39. Aristotle
says (Met. 1074b1) that the ἀρχαῖοι καὶ παμπάλαιοι taught the divinity of the stars ἐν
μύθου σχήματι, but worship of the stars was generally taken as the sign of a barbarian
(Ar. Pax 406, Pl. Crat. 397d). For a certain example of oriental influence on the Pythag-
oreans, see above, ch. II 4, n. 47. Aeschylus calls the stars (Ag. 6) λαμπροὶ δυνάσται
ἐμπρέποντες αἰθέρι.
48 Examples from American Indians to Australian aborigines in J. G. Frazer, The
Golden Bough IV (London, 191 1) 64ff; Capelle 19ff. There are two versions. Fither the
soul of the dead becomes a star, and a shooting star is a soul hastening to its rebirth, or
the living person “has” his star, which at his death is extinguished in the form of a
shooting star. The former version is still influential in Plato’s Phaedrus, as Stenzel showed
(ΚΙ δεν uff; he compares Rep. 621b and Plut. De gen. 22.$91c-d). The latter is the opinion
of the volgus according to Plin. HN2.28: cf. Euseb. Migne 86.1.453, Lucian’s Lychnopolis,
Ver. hist. 1.29, Boll, ZNTW 18 (1917) 4o0ff, Eur. fr. 971. The metamorphosis of the dead
into a star is often mentioned in sepulchral verse (not in prose inscriptions; poetic form
allows more liberty): Peek, nos. 1097, 1829, 1776, 648; Arch. eph. 1953-1954, 2, 290-206.
49 The dead person lives on in the sky, in the retinue of the sun bark (Plut. De Is. et Os.
21; Kees 42, 87ff).—Greek catasterisms go back to a very early period: this is attested
at least for the Bear and Orion (Hes. frr. 148, 163), though these are exceptional cases.
°° Cf. Capelle 3ff; Cumont, Symb. chs. II-III. It is not clear how far back the identifica-
tion of Hecate and the moon-goddess goes (Norden, Vergil VI 23f, and Kerényi 79f
find it as early as Hymn. Hom. Cer. 52), or that of the Gorgoneion with the moon (Meuli,
Bachofen VII 497f).
51 Above, ch. IV 1, n. 47.
52 Ar. Pax 832-837: the slave asks Trygaeus, after his return from the sky, whether
it is true that the dead become stars, as people say. Trygaeus replies that it is indeed true,
and that Ion of Chios, who composed a poem on the morning star, was, after his death,
greeted in the sky as dofos ἀστήρ. This need not be based on anything more than Ion’s
poem; and for the claim that Ion was a “Pythagorean poet” (Cumont, After-Life 95;
Capelle 24f), the fact that he mentions Pythagoras in 2 of the 119 fragments (in Blu-
menthal’s collection) is a slight basis (above, ch. II 3, nn. 13, §1).
360
4. Llarmony of the Spheres and Astral Immortality
of the “divine” stars together with that of the soul to make a proof =
immortality; and Euripides represents Helen as translated a ! ᾿
“palace of Zeus” beyond the starry sky. The idea is often expressec
that man consists of body and soul, and in death the body returns to
the earth, and the soul to the heavenly aether whence it came; in me
epigram for those who fell at Potidaea (432 B.C.) this τ — δ
quasi-official recognition.” It occurs not only in soa ri
Epicharmus; and this might seem to point directly toward a Pyt ἐν
orean origin. But in this respect as in others Euripides is ——
closely, in the tradition, with Anaxagoras.* The latter spoke vn
aetherius calor in the embryo—life comes from the warm aether,
then.5? For Diogenes of Apollonia the soul was a part of the ery
air and thus a μόριον θεοῦ.58 Anaxagoras, too, regarded the soul as
immortal,®® and that is why these two are ranged with Pythagoras.
Medical writers reflect the same belief;8° but even in Leucippus and
58 Above, ch. III 3, n. 97. The etymologies θεός : θεῖν and αἰθήρ : αἰεὶ θεῖν (PI. ie
397¢ 410b) probably related to something earlier—perhaps Alcmaeon, poe a :
Ν Ι ir interpretation o
below, n. 60). If the Pythagoreans, in their in
saa saw motion as the essential characteristic of soul (Arist. De an. aes cf.
above ch II 4, n. 138), both Anaxagoras and Democritus had the same idea (Arist. De an.
403b20ff).
54 Eur. Or. 1683-1690. . ar |
55 IG I? 945 = Peek no. 20 line 5: αἰθὲρ pep φσυχὰς ὑπεδέχσατο, τ Πμα τα δὲ cA
Epicharm. DK 23Bg9: συνεκρίθη καὶ διεκρίθη κἀπῆλθεν oi ἦλθεν πάλιν' ei μ oe
ῦ 7 1ff: daar’... ὅθεν ἕκαστον és τὸ φῶς ἃ ᾿
εἴρια: a ae κοὶ ἐν δὲς yee on δ᾽ és γῆν, Eur. Erechtheus fr. 65.714
ἐνταῦθ᾽ ἀπελθεῖν: πνεῦμα μὲν πρὸς αἰθέρα, τὸ σῶμα δ' ἐς γῆν, Eur. Erechthen 6
eneriee: μὲν οὖν τῶνδ᾽ οὐ BeBao’ “A ιδην πάρα, εἰς δ᾽ αἰθέρ᾽ αὐτῶν πνεῦμ᾽ ἐγὼ Ἔτι τς
Cf. Su 1140, Hel. 1013ff (soul and aether immortal), frr. 839, 877, 971 (also eau "
Psyche Il 25 5f ΙΝ" 435f, Eng. ed.) The same conception on other tomb ier puts: :
11/11F 11466 = Peek no. 1755 (early 4th century B.C.), 12599 (Peek, no. 1759, 3rd century
B.c.); Peek nos. 1760-1761.—Hippoc. Hebd. 52 = Aph. 8.12 (VIII 672 L.).
86 Eur. fr. 839 = Anaxagoras A112, lines 8ff:
χωρεῖ δ᾽ ὀπίσω a
τὰ μὲν ἐκ γαίας φυντ᾽ eis γαῖαν, ᾿
τὰ δ᾽ ἀπ᾽ αἰθερίου βλαστόντα γονῆς
εἰς ὀὐράνιον πάλιν ἦλθε πόλον"
θνήσκει δ᾽ οὐδὲν τῶν γιγνομένων,
διακρινόμενον δ᾽ ἄλλο πρὸς ἄλλου
ἣν ἑτέραν ἀπέδειξεν. " a
87 pee Ατορ. W. H. Friedrich finds the origin of the whole idea in Anaxagoras
(Philologus 97 [1948] 281f).
58 Diogenes of Apollonia A20, 19 $42.
58 A&t, 4.7.1. errr
60 ἐν Hippoc. Carn. 2 (DK 64C3): an ἀθάνατον θερμόν which thinks, sees, hears, =
ἐπο ας τες ἀμρ most of it has “pushed out into the OureEnOs eee τ
this, I think, is what the ancients named αἰθήρ.᾽᾿- --ΤΠε concept of a wore ia ἐν es
sai der Waccden for example, Astr. 25, supposes to be gaa Pyt eee
ics) is implied 1 xctrines, According to Plato (Crat.
the Jonian physics) is implicd in such doctrines ie
Ear tay people believe in a διεξιόν which penetrates the whole world, sometimes
361
me ΔΔΟΣΝΟΝΑΜΥ AND PYTHAGOREANISM
Democritus the soul is “fiery,” like the sun and the moon,” so that
Lucretius, carrying on their materialist tradition, could borrow
expressions from Euripides.” ‘The soul was “fire” according to
Parmenides and Hippasus, but Heraclitus too is said to have called
it scintilla stellaris essentiac.4 And, as carly as Anaximenes, the ψυχή
was related to the “‘divine” air, which surrounds and supports the
cosmos. 58
That the human soul has a very close relationship to the sky and the
stars, anid even that it comes from heaven and returns to it, is thus a
generally held belief in Ionian φυσιολογία, at least from the time of
Heraclitus and Anaxagoras. In the garb of φυσιολογία, and in “ materi-
alist” phraseology,
what starts as a μῦθος continues to exert its influence
sometimes with more emphasis on salvation of the soul, and sometimes
with more on the general thought of microcosm and macrocosm: man
is made of portions of the cosmos, and in death like returns to like.6?
If this whole development were to be traced back to Pythagoras, one
would have to ascribe to his teaching a maximum of influence and a
minimum of definite content; for the testimonia are extremely dis-
parate. The divinity of the stars, which was known to Epicharmus,®®
called sun, sometimes θερμόν, sometimes νοῦς. From the circle of Diogenes come expres-
sions like that of Ar. Nub. 220ff: and Pl. Crat. 396c (the Muse Οὐρανία is ὁρῶσα τὰ ἄνω,
ὅθεν δὴ καί φασιν, τὸν καθαρὸν νοῦν παραγίνεσθαι οἱ μετεωρολόγοι) is not simply
“Pythagorean,” either (pace Boyancé, REG 1941, 156f; Joly 36f).—A “divine” θερμόν
as ψυχῆς δύναμις, ἀνάλογον οὖσα τῷ τῶν ἄστρων στοιχείῳ, is recognized by Aristotle
(Gen. an. 736b29ff, on which see F. Solmsen, JHS 77 [1957] 119-123; H. A. T. Reiche,
Empedocles’ Mixture, Eudoxan Astronomy, and Aristotle’s Connate Pneuma [ Amsterdam,
1960] 97ff; Arist. Περὶ φιλοσοφίας fr. 27 W. = Cic. Acad. 1.26).
1 Leucippus A28, Democr. A 1OI~102; cf. D.L. 9.44.
® Lucr. 2.991ff, after Eur. fr. 8 39.
Ὁ. DK 18.9, 28A45; cf. 46a-b.
*4 Heraclitus Ars: the νοῦς is related to
of the soul πρὸς τὸ ὁμογενές, Stoic inte
here; Heraclitus spoke of the
Kirk in KR 2o5ff).
ὁ Anaximenes fr. 2, and J. Longrigg, Phronesis 9 (1964) 1-4. A θεῖον περιέχον (the
ἄπειρον) is already present in Anaximander.
9“ There is nothing Pythagorean here,” says Nestle (ZN I 610 n. 1) of the Epicharmus
fragments concerned. On the other hand, Pfeiffer (Sterngl. 114) avers that Heraclitus
owed “wichtige Grundziige seiner Weltanschauung” to Pythagoras. It seems to be
Plato's adverse criticism of him that prevents scholars from advancing the same thesis
in relation to Anaxagoras.
“7 Already in the Rig Veda; see H. v. G
1943) 84; Olerud 153. Plato,
to the material.
8 Menander fr. 614 KGrte: ὁ μὲν ᾿Επίχαρμος τοὺς ἰϑεοὺς εἶναι 'λέγει ἀνέμους ὕδωρ
γῆν ἥλιον πῦρ ἀστέρας. To be sure, this sounds less like ancient piety than incipient
skepticism; since one no longer believes in the mythical gods, one restricts himself to
what is visible.
the περιέχον; Ατό §120ff, cf. A17: the return
rpretation and systematization seems to be involved
“death” of the soul (fr. 36), or at least of certain souls (cf.
lasenapp, Die Religionen Indiens (Stuttgart,
Tim. 63¢, purposely restricts this “theory of dissolution”
362
4. Harmony of the Spheres and Astral Immortality
and the analogical conclusion of Alemacon are as far from the idea
that the soul is made of heavenly fire as is the uncomplicated notion that
souls become stars. ΤῸ be sure, these various contradictory ideas are all
at one time or another ascribed to Pythagoras.°° But in Pindar's
exposition of mctempsychosis there is no more trace of astral motifs
than in Empedocles’ theory of the fallen daimon exiled to earth, into
the “cave” (Hades he placed in the realm of air).”° In the monuments
of southern Italian eschatology, there is no reliable, early evidence for
belief in astral immortality.” |
There remains to consider the acusma which asks the ee
“What are the Isles of the Blest?”’ and answers “The sun and moon.
This places the Beyond in the orderly cosmos; it represents the ne
desire for stability that forges a theology of the soul sont of myths
about the soul and puts ritual taboos together into a “way of life
69 Hypomn. 27: The sun, moon, and stars are gods by virtue of the θερμόν wale
in them. The soul is an ἀπόσπασμα αἰθέρος, immortal (28) and invisible (30). After cath
it floats in the air until Hermes conducts the pure souls ἐπὶ τὸν geil Sas ἊΣ sialon
: .v. Hypsistos; Cook, Zeus 76ff;
Cumont, Or. rel. 58, 273 ἢ. 93; REs.v Hy .
st n. 2, is hardly right in seeing Jewish influence here).—Varro in Comm. Bert. Luc.
"491 Usener: “Pythagoras dixit animas in stellas converti virorum fortium (Varro
poe rer. div. fr. 25b Agahd; but see Reinhardt, RE XXII δα ΒΕ aaa jot =
Gon OF aeeal 3 ity i jalogue on the daimonion of Socrates, wh
exposition of astral immortality in the dialog
sui is a Pythagorean circle in Thebes (above, n. 39).—On the moon as oe of souls
see below, n. 75; on the Milky Way and the gates of heaven, below, nn. 90, 94. —
τὸ Cf. ene n. 38, and ch. II 3, n. 80. ἄντρον, Emp. fr. 120; in fr. 6 iene 7 air,
Hera es cf. the Derveni papyrus, col. 18.7. On the reinterpretation of Ν . "3
Cumont Symb. 35ff, Lux 189ff, Nilsson II 228ff. Possible solutions: Hades τὴν ς
night half of the heavenly hemisphere (above, ch. IV 3, n. 64); it sa = qe
and the sublunary region (below, nn. 89, 93); or it could be life on earth ( Pa ᾿ oh
1023; cf. the full treatment of Carcopino, Bas. res ee ee δε ᾿ : ine)
sae i i i docles (A 62), as well as Alem:
from translunary regions is ascribed to Empe Lea
t Empedocles held the
f. above, ch. III 2, n. 32), but we cannot assume that ele
τῶι όσα of the eternal, perfect order of the celestial realm; his cosmos is perishable,
i e “ Sphere.”
d to enter into the perfect unity of the “Sp . ae .
oe Brindisi disc, which depicts the journey of the heey ee in ἂν sare
i i ith refs.), is hard to date. Wuilleumuer decides Οἱ
Nahant yee oh ae ϑο: st. It 5] with other discs
| tury B.c. at latest. It shares |
before the Roman conquest, i.e. the 3rd cen oe
i diac.—The vases of Canosa (3rc
tum a good many motifs, but not the zo . :
ες ροῦν Panik Bachofen made famous have no unambiguously astral oe
(Meuli, Bachofea VII 493ff).—The traces of astral immortality scen in the ¢ ae ates
are sicertain (αὐτὰρ ἐμοὶ γένος οὐράνιον, DK 1B17.7, from Petelia; eae ee
στεφάνου, DK 1818.7, from Thurii, see Carcopino, Bas. 314f, and Diels, we ἐν εἰ
Α Diewrieh KI. Schr. 95 compared Orph. Arg. 761, but in the next verse cree :
‘called θονία βασίλεια).--- Τῆς puzzling conclusion, ἔριφος ets γάλα ἔπετον, wa ᾿ er
i reted ne allusion to the Milky Way by A. Dicterich, De hymnis Orphicts ae ure,
ois 35ff — KI. Schr. 95ff; more emphatically by Carcopino, Bas. oe bal ee
, | julte set religions Il? (Paris, 1909) 125ff; K. $8,
f. Contra, 5. Reinach, Cultes, mrythes et reli Ἴ ᾿ γ53
Die Milch im Kultus der Griechen und Romer (Giessen, 1914) 4160 The word ἔριφος points
to Dionysus (cf. above, n. 45).
72 Jam VP 82 (above, ch. Π 4).
363
ἐν, ASTRONOMY AND PYTHAGOREANISM
(Bios). Height and depth, fall and ascent do not become dominant
ideas in the theology of the soul until the realm of the stars is akan in
to become part of the picture, Related ideas are that the Great and
Little Bear are the hands of Rhea, the Pleidaes the lyre of the Muses
the planets the hounds of Persephone, an earthquake is a conspintonil
mecting of the dead, and that the purpose of thunder is to frighten
those being punished in Tartarus. In this context it is taken for granted
that Hades is beneath the earth, and all the allusions to a Pythagorean
katabasis give the same impression.”? The conclusion that astronom
makes an underground Hades impossible was not drawn by i
Pythagoreans. The Isles of the Blest are not a part of Hades; they are
far removed from the land of the dead. The soul goes there, finally as
to its last resting-place, where it listens, no doubt, to the nen 7
of the “Sirens.” This separation of the Isles of the Blest from ἧς
realm of the dead is doubtless ancient, and presumably the earliest
form of the school’s doctrine, and therefore the total picture, alon
with the acusmata, makes consistent sense. Of course it js wholly ed
astronomical, as is shown even by the simple association of “sun and
moon,” as though they were islands in the same sea. There is no hint
of the multi-storied universe which is standard in the later tradition.”
Here, once more, we find that the acusmata represent a strand of
tradition independent of the later tradition, and also of Empedocles
and Plato,” and evince a Pythagoreanism still innocent of the scientific
view of the world.
Even in Plato’s eschatological myths, in which, according to the
usual opinion, he does most “ Pythagorizing,” astronomical motifs
only gradually make their appearance, showing that he is not repro-
78 Above, ch. II 3 Carcopino, Bas. 2
Ὁ OMe Δ. , Bas. 272 n. 4, sees ἢ ing i sae
by the acusmatici. ere a backsliding into the primitive,
4 Above, nn. 3, 37. There may be some sjonj
12.§2, 167) speaks unmistakably of two sienk Coon te rae ie
*° The moon is very often called the Isle of the Blest (references in Capelle 1off; Cumont
Symb. 177ff; Nock, AJA 1946, 142f), e.g. Castor of Rhodes (ist century B FGrHi é
250F16 Plut. Quaest. Rom. 76.282a, Plut. De fac. 29, Por. ap. Stob. 1.49 δι. ζοῦν pee
6.640, 887). However, the moon is usually merely an intermediate stop, and the ‘ane
ae higher stage (Plut. De fac. 29.944c, Amat. 20.766b, Por. ap. Stob. 1.49.55).
nly in Comm. Bern. Luc. 9.9 (Posidonius according to Reinhardt, RE XXII 589f, 780)
= us pe moon on an equal basis: the soul returns “in suam sedem, hoc est “ solis
a ae ac lunae.” Reinhardt (Kosmos 312 n. 2) doubted the antiquity of the Pythagorean
ΤῸ ΤΕ ἷς difficult to be sure about the relation of the acusma to Alemacon, Empedocle
and Philolaus; one gets the impression that they are harking back to silage aoa
strata—Alcmaeon to the divinity of the stars, Empedocles to the noncosmolo seal tl
of salvation, Philolaus to shamanistic lore about the inhabited moon. ial
364
4. Harmony of the Spheres and Astral Immortality
ducing an already completed system. In the Gorgias, the Beyond is not
brought into any kind of relation to the structure of the cosmos, the
only question is the ethical one, of “what kind of man one ought to
be.” The myth of the Phaedo interprets our life as an existence in the
deeps, by contrast with that much more precious and wondrous life
on high; but in carrying this out, it only describes “the carth,”’ in its
amazing size and remarkable structure.”” The image of the Cave
expresses the same conception of life, but it does not have to do with
the cosmos, but with the νοητὸς τόπος out beyond it.78 In the concluding
myth of the Republic, the image of the spindle with its complicated
spindle whorl revolving in the lap of Necessity, the system of the
planets is depicted, but it has no relationship to the previously de-
scribed “paths of the souls.’ When this description presents us with two
doorways, in sky and earth, with the good souls traveling through the
heaven and the bad ones through the earth, the general conception
presupposed is the pre-scientific one of the flat earth, the hemispherical
heavenly shell, and the dread Underworld. The souls making their
way along the heavenly path naturally find themselves in the company
of the stars, and many have the impression that the stars enter and leave
the sky by two doors.”® An indication of Pythagorean influence on
Plato is the identification of the road “up”’ with that “to the right.’’®°
But the adjustment of scientific and mythical views of the world was
not yet attained.
It is in the Phaedrus that the fate of the soul and the movement of
the heavens are for the first time brought into a really intimate con-
nection. The winged soul-horses, following the godsin their journey to
the heavens, striving, at the zenith of the heavenly vault, for the vision
of the realm beyond the heavens, are borne along in the revolution of
the universe, and brought back, by a circular course, to their starting
point.®! The fall and reascent of the soul are given their most grandiose
7 Pl. Phd. 108d.
78 Pl. Rep. 5170. We need not raise the question here of how much myth and ritual
may lie at the base of the simile of the Cave. (Empedocles has the word ἄντρον in fr,
120.)
79 On the idea of the two heavenly gates, see Cumont, Symb. 4off; Meissner 110.
On Egyptian details (the sun bark, ships of the dead, the gate between upper and lower
world), see Kees 64f, 67ff, 84f.
80 ῬΙ, Rep. 614c; cf. Arist. Cael. 284b6ff.
81 P]. Phdr. 246b: πάντα οὐρανὸν περιπολεῖ, 246C: μετεωροπορεῖ καὶ πάντα τὸν κόσμον
διοικεῖ, 2474: μακάριαι θέαι καὶ διέξοδοι ἐντὸς οὐρανοῦ (διέξοδος is the technical term for
the periodic celestial movements: Hdt. 2.24, Eur. Andr. 1086, Hebd. 1 had ἔξοδος),
2474: ἄκραν ἐπὶ τὴν ὑπουράνιον ἁψῖδα πορεύονται πρὸς ἄναντες, 247ς: αὐτὰς περιάγει
ἡ περιφορά, 2474: ἕως ἂν κύκλῳ ἡ περιφορὰ εἰς ταὐτὸν περιενέγκῃ" ἐν δὲ τῇ περιόδῳ...
λ0ς
ἘΞ Ψ ἜΝ ἂρ αι δ᾽ REV ES OFF ἢ ΤΊ ΔΛ ΚΠΑΙΝΔΙΜ
expression in the Phacdrus. ‘Then comes the ‘Timacus: the Demiurge
creates as many souls as there are stars, for cach star a soul puts oa
Into Its star “as in a wagon,” and thus shows it “ :
universe.” Then it must ee its star, to be ieee pases hc
after a period of trial on earth or another planet it may ΕΠ the
promised return to its σύννομος dorip.*2 Though we hear echoes of
folklore motifs,83 at the same time the relation of soul and celestial
movement has become very close indeed: the form of the world soul
is the moving principle of the cosmos (36b). The same basic ideas are
developed in the Laws, without any mythical garb and with an earnest
claim to embody the truth. The soul, as the principle of self-movement
Ss primary as against any kind of corporeality; soul reveals itself in the
celestial movements, and the scientifically proven regularity and
perfection of these circular movements shows that the soul of the
universe 15 intelligent and good.* Then the Epinomis repeats the same
line of thought, in a systematizing way: the stars are gods, and
astronomy is worship. 88 |
Plato’s students interpreted and systematized their master’s myths
and seem, in this process, to have developed the final form of li
ae Little is known about Xenocrates in this connection. 88
tis a tempting conjecture, thou |
in his ea ἕω On Grief phonies ge ey ree 2
᾿ ght in a popular
way.*’ We can, however, get some impression of the exposition of
Heraclides Ponticus, who also worked on special problems of astron-
omy.** He related the vision of a certain Empedotimus.®® Pluto and
on heme "Ἢ ean influences in the Phaedrus, see K. Kerényi, “‘ Astrologia
; 22 (1923-1924) 2458, Bidez, Eos 60ff; J. Kersch ae
= aa ae ee : : ;J- Kerschensteiner, Platon und
; 3ft; W.J. W. Koster, Le mythe de Plat a
des Chaldéens (Leiden 1951) 4ff. ' : ἀπ leg rainy
5 Chaldé Ξ 4ft.— Even as late as the Phaedrus, the influen ἢ
scientific idea of the world’s structure is felt; the wicked are punished ὑπὸ ee ns But
at the same time the universe
as a perfect sphere, s
therefore no zenith. , Ρ phere, should have no up and down, and
* Tim. 41d et seq.
*§ Namely, the identification of soul
: and star (Stenzel, K/Schr. 8- ab
84 Ρ]͵ Leg. 89Ic, 9674; see also ch. IV 2. ἐνόν,
8δ θερσέβεια Epin. 981d.
86 The myths of Plutarch are suspec ini
ted of contain j
123ff). For the distinction of er regions, and the ees pean any υἱὲ
3 ᾿ : ; : crates
Boyancé, REG 1952, 337ff. Earlier reconstructions of Crantor’s consolatio are critically
analyzed by R. Kassel ae
an κε oh assel, Untersuchungen zur griechischen Konsolationsliteratur (Munich,
i See above, ch. IV 2, n. 11; ch. IV 3, n. 17
ὁ Fer. 93-94. Probably the name is a delib nati
Σ ! : m erate combination of parts of “ Empedocles”
and of “Hermotimus.” In addition to Wehrli’s notes, consult Wikiase CMH I
5338}, Bidez, Eos 52ff. Ancient writers consid i istori
: ’ : ered him a | “al pe ‘fey like En?
menides, Pythagoras, and Empedocles. ma historical personality like Epi-
366
4. Harmony of the Spheres and Astral Immortality
Persephone appeared to him, freed him from the usual restrictions of
human eyesight, and revealed to him the truth about the nature and
fate of the soul. ‘The cosmos, he saw, is divided into three realins, ‘The
spheres of the moon and the elements below it belong to Pluto and
comprise the heavenly Hades. The Milky Way, conceived, as in
Aristotle, as an atmosphcric phenomenon below the region of the
stars,°° is the pathway for these souls; from there they sink to carth,
and later they return to it. The soul is a “light,” αἰθέριον, οὐράνιον σῶμα.
The heavenly Hades is not a place to stay forever; above it are the
spheres of the planets, the realm of Poseidon, and highest of all is the
heaven of the fixed stars, which belongs to Zeus. There must be a
connection between the three realms and the three ‘gates’? which
Empedotimus sees in the region of the zodiac. One is the entrance
Heracles used at his deification.*! Surely, for every soul the ultimate
goal is to follow this road into the company of the gods.
The decisive influence was that of the myths of Plato, Heraclides,
and the other Platonists;* and especially important were the concepts
of the sublunary Hades® and the Milky Way as dwellings of the souls. 4
90 Boyancé, REG 1952, 335 n. 7. The sun is thought of as being below the plancts,
not in their midst (above, ch. IV 1, n. 7). The Milky Way was known to the Babylonians
as a road between earth and heaven (Meissner 111).
91 Fr. 94. The gates are in the sign of Scorpio, between Leo and Cancer, and between
Aquarius and Pisces. Wehrli thinks (92) that this detail is hardly correct, because ‘it
is hard to imagine the zodiac as a path to the subsolar region.” But the zodiac is not a
hollow ring; it pervades all the spheres, and even the moon travels along it. The road
toward incarnation passes through Cancer to Leo, according to Macrobius (Som. Se.
1.12.4, 7), with the sun moving downward, and the first gate is between these two signs.
Pisces and Aquarius are the suitable signs for the realm of Poseidon; and the third gate,
that of Zeus, lies directly opposite, in the middle of the sign itsclf—no longer a transitional
stage. The journey depicted on the disc of Brindisi (above, n. 71) goes to Scorpio, ‘The
division of the circle into portions of 105°, 105°, and 150° (7 : 7: 10) may have seemed
especially “harmonic” to Heraclides (7 is “the rational diameter of 5” in Rep. $46¢).
® This has also been put forth by Boyancé, REG 19$2, 321ff.
8 Ἐκ, Plut. De fac. 28.943c, Sext. Emp. Math. 9.72f, Cornutus §; see above,
n. 75. ὁ
94 Numenius and Cronius ap. Por. De antr. nymph. 28, Macrob. Somn. Se. 1.12, Proc.
In Remp. I] 129, Schol. Od. 13.103 (the discrepancy between this and Porphyry is probably
a mistake of the scholiast rather than a reflection of independent tradition, as Delatte
assumes, Litt. 130). The testimonia are collected by Capelle 39f and Leeman 147f. See
also Delatte, Litt. 129ff. Where the Zodiac crosses the Milky Way, he says, in the sign of
Cancer and Capricorn, there are two gates, one “for men” in the north, through which
souls descend for rebirth, and one “for gods” in the south, through which souls return
to their place of origin. Below the Milky Way begins the realm of Hades; the exposition
is ostensibly an explanation of the Homeric lines about the gate of the sun (Od. 24.12f)
and about the cave of the Nymphs (Od. 13.102ff). The only part of this that is explicitly
attributed to Pythagoras is that the souls which are punished with reincarnation gather in
the Milky Way; and this is brought into connection with the fhet that the infant's first
food is milk. On the Milky Way as a habitation of souls see also Cic. Rep. 6.13, 16;
Gundel, RE VII 564-5066; cf. above, n. go.
367
Γι, ASTRONOMY AND PYTIHAGOREANISM
Probably Heraclides mentioned Pythagoras, too, in his Empedotimus,*
but a simple inference from the ‘Timaens may be sufficient cause why
the details of astral immortality in later tradition were attributed to
Pythagoras. These testimonia do not have pre-Platonic content. Of
course the interpretation of the Milky Way could be combined with
the naive identification of soul and star, but the Pythagoreans of
Aristotle's accounts had a different explanation of the Milky Way.
The idea of two heavenly gates is simpler and therefore earlier than
the picture drawn by Heraclides, but they belong primarily to a
different realm.%8
Thus the idea of astral immortality only gradually developed into a
system apparently built on a scientific basis. At the inception stands
shamanistic “knowledge” of cosmos and soul, in the most ancient
stratum of the Pythagorean tradition, the acusmata. Later on, scientific
discoverics became current. The knowledge that the earth is spherical
did away with the subterranean Hades—from the time of Empedocles.
Recognition of the orderly character of the movements of the planets
confirmed—from the time of Eudoxus—the contrast of celestial order
and earthly imperfection. This was the path that led to that synthesis
of astronomy and religion which we find in the later Plato, in Hera-
clides, Aristotle, and Xenocrates: we cannot simply call it “ Pythag-
orean.”’ This doctrine then, taking its departure from Plato and
Aristotle, finally became canonical. The agreement of science and
religion, cmphasized by the Stoics, obviously made a tremendous
pression on the Romans. That it was, basically, a hasty over-
simplification remained unnoticed until, from the time of the Renais-
sance, Greck natural science was carried forward by new methods.
" Julian writes (ap. Suda s.v. ᾿Εμπεδότιμος, Heraclides fr. 92): ἡμεῖς δὲ ᾿Εμπεδοτίμῳ
καὶ ΠΠυθαγόρᾳ πιστεύοντες οἷς τε ἐκεῖθεν λαβὼν ᾿Ηρακλείδης ὁ Ποντικὸς ἔφη... But
Julian is not citing at first hand (Wehrli 91). Wilamowitz says of “Pythagoras’”
teaching about the Milky Way, ‘Pythagoras has taken the credit here that belongs
to Heraclides” (GldH I $35 n. 1).
™ Above, n. 48. The conception of the Milky Way as a collection of many single stars
does not seem to be documented, however, before Democritus (Ag1).
* Above, ch. IVo1, m ris.
™ Above, n. 70. Perhaps the two gates were assigned to the zodiac (Cancer and Capri-
corn) even before Heraclides, as Wehrli supposes (92); Bidez (Eos 286 thinks of Eudoxus
in this connection,
368
V. Pythagorean Musical Theory
I. SPECULATION, EXPERIMENTATION, AND FICTION
It is a striking paradox that music, which is the most spontaneous
expression of psychic activity, at the same time admits, or rather even
challenges, the most rigorous mathematical analysis. There are two
systems by which the division of the tonal continuum may be des-
cribed, and the distinction between natural and tempered tuning
follows from them. One can think of the interval in a spatial metaphor,
and equal intervals as representing equal distances or lines; then greater
intervals are made up of the sums of smaller ones. The standard unit
is the “tone” (or “‘step’’), the difference between a fourth and a fifth;
it can be subdivided at will. A fourth, then, comprehends 23 whole
tones, a fifth 34, and an octave six. The usual divisions of the tetrachord
in classical Greek music are, in the diatonic genus or scale, semitone,
whole tone, whole tone; in the chromatic, semitone, semitone, tone
and a half; in the enharmonic, quarter-tone, quarter-tone, ditone.
The image of a line and its divisions is especially natural for us, because
of our familiarity with the piano keyboard, and of our system of
musical notation; but the Greeks used this image, too, as is shown even
by the word they use for “interval,” διάστημα." |
A different system results from the recognition that the harmonic
intervals can be expressed as simple numerical ratios. These can easily
be illustrated by the length of vibrating strings or sounding pipes. It
has been known for a long time that pitch depends on the rate of
vibration; and simple, whole-number ratios of frequencies result in
the musical concords. The ratio for the octave is 2:1, for the fifth,
3: 2, for the fourth, 4: 3. Addition of intervals results in multiplication,
and subtraction results in division, of the numerical ratios; to halve an
. interval means the extraction of a square root. In the terminology of
modern mathematics, the intervals, thought of as lengths of line,
1 According to modern usage, the scales are here considered as ascending series. i
the following we shall employ the usual transcription, related to the modern system?
notation (the absolute pitch being optional). Thus the tetrachords are ὁ fg a (diatonic),
e fg flat a (chromatic), ande οὐ f(- g double-flat) a (enharmonic).
349
SMe 402 ΦΔΙΚΥΝΔΜΑΥΣΣ ORES ΕΎΊΠΙΑΛ ΚΠ ΙΝ ΜΙ
Probably Herachides mentioned Pythayoras, too, in his Empedotimus,®
but a simple inference from the ‘Timacus may be sufficient cause why
the details of astral immortality in later tradition were attributed to
Pythagoras, These testimonia do not have pre-Platonic content. Of
course the interpretation of the Milky Way could be combined with
the naive identification of soul and star,°* but the Pythagoreans of
Aristotle's accounts had a different explanation of the Milky Way.*?
The idea of two heavenly gates is simpler and therefore earlier than
the picture drawn by Heraclides, but they belong primarily to a
different realm.%8
Thus the idea of astral immortality only gradually developed into a
system apparently built on a scientific basis. At the inception stands
shamanistic “knowledge” of cosmos and soul, in the most ancient
stratum of the Pythagorean tradition, the acusmata. Later on, scientific
discoveries became current. The knowledge that the earth is spherical
did away with the subterranean Hades—from the time of Empedocles.
Recognition of the orderly character of the movements of the planets
confirmed—from the time of Eudoxus—the contrast of celestial order
and earthly imperfection. This was the path that led to that synthesis
of astronomy and religion which we find in the later Plato, in Hera-
clides, Aristotle, and Xenocrates; we cannot simply call it “Pythag-
orean.” This doctrine then, taking its departure from Plato and
Aristotle, finally became canonical. The agreement of science and
religion, emphasized by the Stoics, obviously made a tremendous
impression on the Romans. That it was, basically, a hasty over-
simplification remained unnoticed until, from the time of the Renais-
sance, Greek natural science was carried forward by new methods.
* Julian writes (ap. Suda s.v. ᾿Εμπεδότιμος, Heraclides fr. 92): ἡμεῖς δὲ ᾿Εμπεδοτίμῳ
καὶ Π]υθαγόρᾳ πιστεύοντες οἷς τε ἐκεῖθεν λαβὼν ᾿Ηρακλείδης ὁ Ποντικὸς ἔφη... But
Julian is not citing at first hand (Wehrli 91). Wilamowitz says of “Pythagoras’”
teaching about the Milky Way, “Pythagoras has taken the credit here that belongs
to Heraclides” (GidH II $35 n. 1). :
*° Above, n. 48. The conception of the Milky Way as a collection of many single stars
does not seem to be documented, however, before Democritus (AgI).
ὅτ Above, ch. IV 1, n. 115.
** Above, ἢ. 79. Perhaps the two gates were assigned to the zodiac (Cancer and Capri-
corn) even before Heraclides, as Wehrli supposes (92); Bidez (Eos 28ff) thinks of Eudoxus
in this connection.
368
V. Pythagorean Musical: tneory
I. SPECULATION, EXPERIMENTATION, AND FICTION
It is a striking paradox that music, which is the most spontanco
expression of psychic activity, at the same time admits, or rather evi
challenges, the most rigorous mathematical analysis. There are tw
systems by which the division of the tonal continuum may be de;
cribed, and the distinction between natural and tempered tunin
follows from them. One can think of the interval in a spatial metaphon
and equal intervals as representing equal distances or lines; then greater
intervals are made up of the sums of smaller ones. The standard unit
is the “‘tone” (or “‘step’’), the difference between a fourth and a fifth;
it can be subdivided at will. A fourth, then, comprehends 2} whole
tones, a fifth 34, and an octave six. The usual divisions of the tetrachord
in classical Greek music are, in the diatonic genus or scale, semitone,
whole tone, whole tone; in the chromatic, semitone, semitone, tone
and a half; in the enharmonic, quarter-tone, quarter-tonc, ditone.
The image of a line and its divisions is especially natural for us, because
of our familiarity with the piano keyboard, and of our system of
musical notation; but the Greeks used this image, too, as is shown even
by the word they use for “interval,” διάστημα."
A different system results from the recognition that the harmonic
intervals can be expressed as simple numerical ratios. These can easily
be illustrated by the length of vibrating strings or sounding pipes. It
has been known for a long time that pitch depends on the rate of
vibration; and simple, whole-number ratios of frequencies result in
the musical concords. The ratio for the octave is 2:1, for the fifth,
3: 2, for the fourth, 4: 3. Addition of intervals results in multiplication,
and subtraction results in division, of the numcrical ratios; to halve an
interval means the extraction of a square root. In the terminology of
modern mathematics, the intervals, thought of as lengths of line,
1 According to modern usage, the scales are here considered as ascending series. In
the following we shall employ the usual transcription, related to the modern system) of
notation (the absolute pitch being optional). Thus the tetrachords are ὁ f g a (diatonic),
e fg flat a (chromatic), andee* ἔ{- g double-flat) a (enharmonic).
349
EE NESE SEY SEU UILAL LALOR Y
correspond to the logarithms of the respective ratios.2 To this extent
the two descriptive systems are equally accurate and can be converted
into cach other. [Ip antiquity this mathematical schematism was un-
known, and there was no adjustment of the two systems to cach other.
In particular, irrationality was excluded from the theory of musical
proportion. The line-segment idea is connected with the name of
Aristoxenus, the theory of musical Proportion with that of the
Pythagoreans.5 The two schools, Aristoxenians and Pythagoreans,
stood in a relation of hostility: the latter accused Aristoxenus’ followers
2So, for example, fifth + fourth = Octave, ie. (3: 2) Χ (4:3)=2: ι; “fitth: =:
fourth = whole tone, 1.6., (3:2): (4 13) = 9:8. The “semitone” that remains in the
diatonic tetrachord, when two whole tones are subtracted from the fourth, is (4: 3):
(9:8): (9: 8) = 256: 243, the so-called leimma, somewhat smaller than a tempered
semitone (V9: 8) (Pl. Tim. 36b). Ancient theoreticians usually went through a somewhat
more laborious Procedure, involving a step-by-step extension of the Proportions; e.g.,
the fifth is 3:2 — 9:6; the fourth is 4: 3 = 8:6; so that the whole tone is 9: 8 (Theo
Sm. 67.16ff, etc.). The fundamental importance of the concept of linear distance is
and division, was the point of departure for the development by Napier of the method
of calculation by logarithms, “Log-arithm” means “‘ratio-number” (Tannery, MSc Il]
71).
3 For the method of conversion, and comparison of the values, especially, of Aristoxenus
and Archytas, see Tannery MSc Ill 98. Of course, equal division of the interval-“line”’
leads to ratios of vibration frequency that are almost exclusively irrational (“tempered
tuning”’), whereas frequency ratios using whole numbers lead to transcendental division
of the interval-“line” (“natural tuning”’), making impossible the establishment of ἃ
the pitch (below, n. 41) seemed to exclude irrationality: we are told that κινήσεις
πυκνότεραι OF ἀραιότεραι consist of “ Parts’ and therefore must be related like whole
numbers, Sect. can. prooem. 158.7ff Menge. This overlooks the fact that it is not the num-
ber of vibrations itself which is in question, but the number per time unit. The inter-
polation of the arithmetic and harmonic means leads directly to approximative values
for the square root (Tannery, MSc Ill 83), but it is unlikely that the study of irra-
*Didymus wrote περὶ διαφορᾶς τῆς Πυθαγορείου μουσικῆς πρὸς τὴν ᾿Αριστοξένειον
(Por. In Ptol. 5-11f, 25.5f, cf. 3.13f; fragments: 26.6ff, 27.1 -7fF, 107.1 58} it was used without
recognition by Ptolemy—see the criticism, Harm. Ρ- 14.1ff, of what Por. reports at
370
t Speculation, Experimentation, and Viction
of imprecision, and were accused, in their turn, of using ee
hypotheses and contradicting the clear testimony os the ise 4
In the one passage where he explicitly names of Πυθαγόρειοι, | lato
credits the Pythagorcans with a mathematical theory of music;? this
is one of the few fixed points in the reconstruction of Pythagorcanism
before Plato. As early as Xenocrates, the crucial discovery was attribu-
ted to Pythagoras himself; and though this testimony is treated with
great reserve, still it is generally regarded as established that the first
natural law to be formulated mathematically—the relation between
pitch and the length of a vibrating string—was a discovery of the
Pythagorean school. Let us try, however, to answer somewhat more
precisely the question, to what extent, and from what time, there was
anything in this area truly analogous to modern science. For in this
field, as in others, it may be that a different kind of speculation preceded
real science.
In the Republic, when he discusses the necessary subjects for the
education of the “Guardians,” Plato takes up music after astronomy;
they are “sister sciences, as the Pythagoreans say and we agree
ἀδελφαί τινες αἱ ἐπιστῆμαι... ὡς of τε Π]υθαγόρειοί φασι καὶ ἡμεῖς...
συγχωροῦμεν (530d). Socrates will follow them, he says,® but they do
not fulfill adequately the basic requirement, to push forward beyond
the sensible world to true Being and thus to really exact knowledge:
“they waste their time like the astronomers, measuring audible con-
cords and sounds against one another” (5314).
Glaucon, the interlocutor, follows this up with a depiction of the
activities of the musicologists, bending over their strings and arguing
whether it is possible to detect still another difference of pitch, or
whether the “smallest interval, the unit of measurement” has been
reached." These “worthy men,” says Socrates mockingly, subject the
strings to a painful inquisition, stretching them on the rack, with pegs.
But, he continues, these are not the musical theorists he meant, but
® According to Ptolemais of Cyrene (Por. In Ptol. 23.2 5ff) the ae as ὑπ
if the finding of theory (Aéyos) contradicted that of the senses (atoByos), the latter was
false; ‘‘ This is the canonists’ way of thinking, carried in absurdum (Diiring 1934, 144).
7 Rep. 30d et seq.; also Arist. - ater et sacp., Eudemus fr. 142.
: 5 fr. 9; cf. above, ch. I 3, n. 68.
Sane $30d. On the passage as a whole, sce Frank 1 Soff, IK. Moutso~
poulos, La musique dans Voeuvre de Platon (Paris, O59) 48fF; Richter $4ff
10 $31a: τὸ σμικρότατον διάστημα, ᾧ μετρητέον.
371
-«--
cia a ἐς ἐς. Ὁ
Γ΄ PYTHAGOREAN MUSICAL ‘THEORY
those of whom we were just saying that we would consult them
, Δ] ᾿
about harmony”—that is, the Pythagoreans (5310). They, too,
however, fall into an error similar to that of the astronomers:
They try to find the numerical properties hidden in these audible
consonances, but they do not rise to the level of formulating prob-
lems and investigate which numbers are consonant and which not
and why! |
Thus Plato distinguishes two schools of musical theorists. One group
try to find the smallest interval, as a unit of measurement, proceed in
a purcly empirical way, and rely on their musical sense. They must
conccive of the tone continuum as a line, since they are seeking the
basic unit, and are therefore the predecessors of Aristoxcnus.2 Their
method of study is, to Plato, not even worth discussing; but for the
Pythagoreans, who are looking for the “numbers” in the musical
concords, he has appreciation to mix with his criticism. He will follow
them in detail, though they are still too closely wedded to the empirical.
What Plato desiderates is not an analysis of audible music but pure
number theory, above and beyond experience. In the Timaeus, Plato
carried out this program, at least by way of suggestion, using a serics
of numbers derived from the ultimate principles, which arrayed them-
selves in a scale without audible sound, the numerically harmonic
structural pattern of the world, the “world soul.’23 Succeeding ages
™ 5310. πρόβλημα is a technical term in mathematics: see Oenopides DK 41.12
(τίνος ὄντος τί ἐστιν), Pl. Rep. 530b: προβλήμασιν... χρώμενοι ὥσπερ γεωμετρίαν
οὕτω καὶ ἀστρονομίαν μέτιμεν.
"ἢ Among non-Pythagorean music theorists we can name the following: Lasus of
Hermione, the teacher of Pindar, was the first to write περὶ μουσικῆς (Suda s.v., Mart
Cap. 9.936; see Schmid—Stihlin 1 1, 544ff ). Epigonus (Aristox. Harm. 1 p. 3.21, Philochorus
Gr Hist 328F23 -= Ath. 14.637f, Ath. 4.183d, Por. In Ptol. 3.4) constructed an instrument
with 40 strings (Pollux 4.59). Stratonicus, according to Phaenias fr. 32 W., was the first
to make a διάγραμμα (on his chronology, sce above, ch. II 4, ἢ. 43). Then there are also
Eratocles (Aristox. 1 p. 5.9ff, Por. In Ptol. 3.5), Agenor (Aristox. Harm. 2 Ρ. 37, Isoc
Ep. 8.1, Por, In Ptol. 3.5), and Pythagoras of Zacynthus (Ath. 14.637bff, Aristox. Fat Σ
Ρ. 36f). In all these the combination of practical and theoretical music js characteristic
On the other hand, Damon (IDK 37) is only cited for the ethical and pedagogical value οἵ
πμμδίσς That Aristotle, too, knew two schools of musical theorists is shown by his
expressions of κατὰ τοὺς ἀριθμοὺς ἁρμονικοί (Top. ' ᾿ ἡ ἐ is
ena e gue ριθι ppovixot (Top. to7ars) and ἡ ἐν τοῖς μαθήμασιν
The relationship of the Tinacus and the Republic on this point was emphasized hy
Frank, 13ff, ruth See also Rivaud, Rev. Hist. Philos. 3 (1020) HA On the interpretation
of the number series, see Frank 163f, and more recently 1}, Kytzler, “Die Weltseele und
der musikalische Raum’, Hermes 87 (1959) 393 413. After the “One” (which does noe
countas a number, Euclid 7 dets, 1-2) come the 2 first numbers 2 and 3 (exemplars of
“even” and “odd; Adrastus ap. Theo Sin, 94.124). They are raised to the third power
372
Ι. Speculation, Experimentation, and Liction
regarded the construction of the world soul in the Timaeus as one of
the most illustrious examples of Plato’s “Pythagorean” wisdom; but
his own words, in the Republic, show that he went beyond the teach-
ings of the Pythagoreans in an independent way.
Frank tries to determine the character and chronological position of
Pythagorean musical theory from Plato’s pronouncement: he “‘rejects
their comparative measurements as the most contemptible empiri-
cism.’4 “What these scholars are striving for is, in a word, natural
science and physics on a mathematical basis, quite in our modern sense”
(172). It was Plato who first introduced a priori speculations into music
theory; and this is another reason, he thinks, to regard the Philolaus
fragments as spurious. The time of origin of Pythagorean music theory
is “established beyond question” (159) by Plato: Glaucon’s misunder-
standing showed (Frank thought) that the Pythagorean theory was
practically unknown in Athens in Socrates’ time, and had therefore
only emerged about 400 B.c., in the circle of Archytas. Of course, the
numerical ratios that make the basic concordant intervals must have
been known, on an empirical basis, for a long time—“‘every maker
of instruments had to know these numerical formulae” (11); but it
was only the theory of proportion, on one hand,! and on the other
the recognition of the nature of sound as air vibrations, that made
possible the origin of Pythagorean musical theory.
Van der Waerden paints a different picture, on the basis of a much
more even-handed study of the sources. Pythagoras himself, he thinks,
may be credited with the arithmetical manipulation of the basic
harmonic ratios, which had been known for a long time, and also with
the recognition that sound is derived from movements of air16 But
even in Archytas, in his view, considerations of number theory, rather
because it is the three-dimensional physical world which is to be constructed (cf. τρὶς
avénbeis Rep. 546c, Arist. Pol. 1316a7f, Pl. Tim. 32b [in a proportion with 2 mean
proportionals, the outer members are cubes], Leg. 8944 and the succession line—plane-
solid [above, ch. 1 1], Epin. 991a, and the ancient commentators on the Timaeus, Adrastus
ap. Theo Sm. 65.1ff, 95.21ff, and Procl. In Tim. ΠΕ 170). The further rule, to supplement
the resulting ἐπίτριτα with ἐπόγδοα (36b), cannot be explained by a priori considerations,
but only from the needs of music.
14 Frank 152, cf. 13; I6I, 172.
15 Frank’s statement that ‘‘the proportion theory of the intervals was first worked out
by Eudoxus”’ (160) is not supported by Theo Sm. 61.11ff (= Archytas A1tga); Eudoxus
made the theory of proportion applicable to irrational relationships (cf., e.g., Becker,
music theory (above, n. 4); the bases of calculation by ratios are much older; cf. ch. ΝΠ 1.
18 Herntes 1943, 179, 192. Van der Waerden rightly emphasizes the age of the aciusmata
tradition. (Above, ch. Il 4, n. 1§7 on the tetractys.)
τ ΙΝ ἈΝ EOENENAENEMEEV νει} 31,31}, ΣΡΙΠΒΟΏΚΥ
than empirically exact measurements, were most important; and the
monochord was probably not invented until after Archytas’ time.
Thus the course of the development would be, from the everyday
experience of the maker of instruments to the theories of the Pytha-
gorcans, and then to the experiments that can really be called scientific.
Van der Waerden did not take accqunt, however, of the fact that
alongside, and prior to, the Pythagorean theory of music there was
also a non-Pythagorean music theory and a non-Pythagorean natural
science.
The question of what observations lay behind the discovery of the
numerical relations of musical intervals, and of when this happened,
is harder to answer than it first seems. The often expressed opinion,
that these numbers had been known “from time immemorial” in the
daily work of the makers of musical instruments,!” does not comport
well with the nature of Greek instruments. The most common stringed
instruments have strings of equal length and no finger board, though
flageolet tones seem to have been recognized,'8 In the triangular harp
the tension of the strings and their thickness played some role, but we
do not know Just what.!9 In a wind instrument with finger holes, that
is, the aulos, the distances between the holes do not correspond directly
or accurately to the ratios of the intervals ;20 actually the holes were
simply bored at equal distances. The syrinx was not used in the music
of the classical period.2! And the monochord with a movable bridge,
the κανών, the only “instrument” on which Pythagorean musical
1” So Frank r1f, 161; Tannery, MSc III 241; Kranz, Philologus 1938, 437; van der
Waerden, Hermes 1943, 172.
18 διάληψις Arist. Pr. 19.12 cf. 23; Theo Sm. 59.22; Jan, Musici scriptores graeci p. 84 n.
*° A reference to the τρίγωνα ψαλτήρια τῆς ἴσης ἐπιτάσεως γινομένης (ἢ Arist. Probl.
19.23. M. Schmidt, Zur Entstehung und Terminologie der elementaren Mathematik (Leipzig,
1914”) 108, derives the discovery of the numerical relationships from this “Egyptian
harp”; similarly Farmer ap. E. Wellesz (ed.), New Oxford History of Music 1 (Oxford,
1957) 275. Cf. Ael. ap. Por. In Ptol. 34.20ff. ,
20 Ptol. Harm. 1.8, p. 17.2ff; 2.12, p. 66.31: flutemakers (of τὰ ἔμπνευστα ὄργανα
κατασκευαζόμενοι) proceed in a completely empirical fashion. The assertion that they
placed the finger holes with reference to the numerical ratios is a product of the scholarly
imagination (Arist. Pr. 19.23: καὶ of αὐλοτρῦπαι οὕτω λαμβάνουσιν, Ael. ap. Por. In
Ptol. 34.21ff, Theo Sm. 61.2, Nicom. Ench. 6 p. 248.15, 10 p. 255.4ff, Por. In Ptol.
ss ). Contra, Aristox. Harm. 2 P. 37.25ff; K. Schlesinger, The Greek Aulos (London,
1939).
"1 It only came into common use as a musical instrument with the water organ (Por.
In Ptol. 119.28f, ὕδρας). References to experiments with the syrinx: Arist. Pr. 19.23,
Theo Sm. 60.6f, Cens. 10.10f, Nicom. Ἐπεὶ. 6 P. 248.15, Por. In Ptol. 119.14ff. The
remoteness of the theorists from practical music is shown in the nonsensical statement
that the pipes had to be equal in thickness, and that wider tubes would produce higher
tones (Ael. ap. Por. In Ptol, 34.11ff).
374
1. Speculation, Experimentation, and Fiction
theory can be demonstrated with any approach to exactitude, is an
artificial device for experimentation, the time of whose invention is
controversial,#
An extensive and richly attested tradition makes Pythagoras himself
the discoverer of the numerical ratios: but the observations and
experiments attributed to him are impossible, physically. One story is
that in passing a smithy he recognized, to his surprise, that the sounds
made by the hammers exemplified the intervals of fourth, fifth, and
octave. He ascertained that the only difference among the hammers
was their weight, and found that their weights were related in the
ratios 4: 3, 3:2, and 2:1. The law presupposed here, that the vibra-
tion and sound of a metallic body are directly proportional to their
volume and weight, is false.28 The story goes on that Pythagoras hung
weights corresponding to these hammers from equally long strings,
and found, on plucking them, that the same intervals were produced.
2 The crucial point is the Sectio canonis (Κατατομὴ κανόνος), transmitted under the
name of Euclid (ed. H. Menge). Euclid’s authorship is contested (Tannery, MSc III 213ff;
Menge xxxvii et seq., hesitantly During 1934, 177), and in addition Tannery claimed that
the only passage to mention the κανών (props. 19-20) is an addition made in the time of
Eratosthenes. In these propositions the diatonic scale is constructed, but in prop. 17 the
λιχανός is 2 whole tones below the Mese, which presupposes the enharmonic scale. But
aside from the fact that 19-20 may have displaced an older, enharmonic sectio canonis,
the juxtaposition of enharmonic and diatonic is comprehensible; The enharmonic is
basic to musical practice, and hence comes the name λιχανός = f; but as the basis for
μεταβολαί one needs the diatonic ἀμετάβολον σύστημα, which is built up in props.
19-20, and here we do not find the word λιχανός but only διάτονος =: g. That the
illustrative figure gives λιχανός means nothing, in view of the uncertain transmission
of such drawings. Van der Waerden (Hermes 1943, 172ff, 177) therefore puts the invention
of the κανών after 300 B.c.; but, since neither Tannery’s analysis nor the conclusion
drawn from it is certain, we may go back further than that. Aristotle never mentions the
κανών; for him music belongs to arithmetic, just as optics does to geometry (An. post.
75b14ff, 76a9ff, 22ff, Met. 1078ar4ff), while the κανών presents a combination of music
and geometrical line-division. We should probably be justified in taking this as a terminus
post quem. H. Koller dates the invention of the κανών in the sth century (Glotta 38
[1959] 668), but his derivation of logic and epistemology from music theory is pure
construction. Duris speaks of the κανών (FGrHist 76F23); see also Philodemus (Mus.
p- 100 Κι) and Varro (Gell. 16.18.4). Its invention is attributed to Pythagoras by 1, L,
8.12, Gaudentius 11 p. 341.12ff Jan, Boeth. Mus. 1.11; cf. Aristid. Quint. 3 p. 116 M.,,
Procl. In Tim. 11 174.23, Por. In Ptol. 120.17ff. We can disregard here the other experi-
mental devices that Ptolemy describes (Harm. 2.2, 2.12f, 3.1). See also H. Oppel, KANQN
(Leipzig, 1937; Philologus Supp. 30.4). Late texts on the “ Pythagorean kanon” have been
edited by A. Stamm, Tres canones harmonici (Diss. Strassburg, 1881).
*8 The physical impossibility of the alleged experiments of Pythagoras was shown by
M. Mersenne, Questions harmoniques (Paris, 1634) 166 (Schuhl, Essai 262.2; Capparelli ἢ
627). See also van der Waerden, Hermes 1943, 170ff; H. Oppermann, “Eine Pythagoras-
legende,” Bonn. Jb. 130 (1925) 284-301. For the narrations, see Nicom. Ench. 6 Ρ. 244
= Tam. VP 115ff, abbreviated Iam. In Nic. 121.13ff, most vividly Macrob. Somn. Sc.
2.1.9ff; also Gaudentius 11 p. 340 Jan, Boeth. Mus. 1.10, Isid. Ht. 3.16.1 (cf. Zeller I
$08.1).
3274
Fe HS ἃ δάδδιξι ANGIE EEN EVER ESTs ἢ TEGAN SAIN FE
But again, the proposition that the frequency of vibration of a string
is proportional to its tension is false."4
Our oldest attestation for this tradition is that of Nicomachus and
Adrastus. It is definitely earlier than Ptolemy, who rejects the weight
experiment, since it could only bring διαβολαί to the correct theory.”
Thus the inauthenticity of the experiments had already been recognized
and used in polemic against the Pythagorean musical theory. It is
unlikely that Adrastus and Ptolemy are dependent on Nicomachus,?*
for the exegesis of the Timaeus reflected in Adrastus and Macrobius is
based on earlicr material. In fact, Xenocrates attributed the discovery
of the musical ratios to Pythagoras himself (above, n. 8); and it is as
unlikely that he omitted to mention the way the discovery was made
as it is that he mentioned the κανών (above, n. 22). Should we suppose
that here again we have a falsified tradition about Pythagoras emerging
from the Old Academy?
In any case the legend, in spite of its physical impossibilities, does
make a certain kind of sense. The mythical inventors of smithcraft, the
Idacan Dactyls, were regarded not only as wizards and founders of
mystic rites, but also as the inventors of music. They were mentioned
along with the Curetes and Corybantes, but also with Orpheus, and
even with Pythagoras.?” Music magic is found throughout the mystery
cults’® and takes on a special character among the Pythagoreans.
Because of this ritual and magical background, we should take
scriously Aristoxenus’ reports about the Pythagoreans’ musical
κάθαρσις .39 and Plato’s conception of music belongs in this context.2°
The acusma which states that the sound of bronze when struck is the
"ὁ Nicomachus, Iamblichus, Gaudentius, Macrobius, Boethius, as cited in the preceding
note; without the story of the hammers, Adrastus ap. Theo Sm. 57.4 = Chalcid. 45,
Cens. 10 (Varro?), Por. In Ptol. 119.29ff, cf. Theo Sm. 60.7f; 66.21ff, Aristid. Quint. 3
p. 113 M. For an attempt to explain why such an experiment fails to work, see Ptol.
Harm. 1.8 p. 17.7ff. In reality, the pitch or frequency is proportional to the square root of
the tension; e.g., 4 times as much weight produces an octave; but this law does not seem
to have been discovered in ancient times (Tannery, MSc III 440). The κανών is mentioned .
last in Porphyry, Gaudentius, and Boethius; and, in Nicom. Ench. 10, it is treated without
any mention of the discovery of musical ratios.
25 Ptol. Harm. p. 16.32ff.
“Ὁ Adrastus was about contemporary with Nicomachus. He does not tell the impressive
story of the smithy.
*” See Kern, RE IV 2018ff; B. Hemberg, Eranos 50 (1952) 41-59; esp. Ephorus FGrHist
7OF104 = Diod. 5.64, Plut. Mus. §, Clem. Al. Strom. 1.73.1, Solinus 11.6. Terpander was
regarded as the descendant of the Dactyls, Schol. T. 1]. 22.391, Pythagoras as an initiate
of the Dactyls, Por. VP 17.
8 Boyancé, Muses passim.
® Aristox. frr. 26, 121; also lam. VP 64ff, r10ff, Aristid. Quint. 2 p. 110 M., Schol.
T Hi. 22.391, Por. VP 30, 32f, lam. VP 163f, 224. See also Zeller I 406.2; Boyancé, Muses
93ff, Rostagni, ScrMin I 135ff. The music of the monochord accompanied the death of
Pythagoras, Aristid. Quint. 2 p. 116 M.
30 Tim. 47d; Boyancé, Muses 173.
376
ΒΒ | ἐπ ial bi hata ie ΗΝ ec liad eee Se ee ee ee ee
voice of a daimon™ makes the transition, in the Pythagorean milieu,
between music and metal-eworking. The claim that Pythagoras dis-
covered the basic law of acoustics in a smithy is a rationalization ---
magical music which was discovered by the mythical blacksmiths.®
A report about Hippasus, based on Aristoxenus, is of a different
kind: “‘Hippasus prepared four bronze discs in such a way that their
diameters were equal, while the thickness of one was $ that of the
second, 2 that of the third, and double that of the fourth; when struck,
they made concordant intervals.’’*? ‘This experiment is “correct,” in
terms of the physical principles involved. With free-swinging circular
metal plates of the same diameter, the vibration frequencies are
directly proportional to their thickness.5* Therefore we must regard
as authentic the statement that Hippasus knew and studied the numeri-
cal ratios of the basic concords.®°
Another report associates Lasus of Hermione with Hippasus:
ε ‘ ᾿ ~
Λᾶσος δὲ ὁ ‘Eppioveds, ὥς φασι, καὶ ot περὶ tov Meramovrivov
Ad \ Mv 7 8 ἴω , 4 fa
Ϊππασον ΠΠυθαγορικὸν ἄνδρα συνέπεσθαι τῶν κινήσεων τὰ τάχη
- τ € / 3 3 - ξ ,ὔ
καὶ τὰς βραδυτῆτας, δι᾿ ὧν αἱ συμφωνίαι «.. .» ἐν ἀριθμοῖς ἡγούμενος
> > 3 ἤ ” A 3 ᾿ ¢ i
λόγους τοιούτους ἐλάμβανεν ἐπ᾽ ἀγγείων. ἴσων yap ὄντων καὶ ὁμοίων
“-"ἍΟ ᾿ 2. 4 1 @ ε ΄-Ἀ ’ὔ
πάντων τῶν ἀγγείων τὸ μὲν κενὸν ἐάσας, τὸ δὲ ἥμισυ ὑγροῦ πληρώσας
~ ‘ ~ 9 id ’
ἐψόφει ἑκατέρῳ, καὶ αὐτῷ ἡ διὰ πασῶν ἀπεδίδοτο συμφωνία...
The text can scarcely be sound,®* but Lasus seems to be the subject
throughout, as well as in the succeeding passage about the subdivision
31 Above, ch. II 4, n. 34.
32 Burnet, EGP 106f, said of these stories, “‘ Their absurdity is their chief merit. They
are not stories which any Greek mathematician could possibly have invented, but popular
tales bearing witness to the existence of a real tradition that Pythagoras was the author of
this momentous discovery.” But the content of the tradition changed as it was rationalized.
38 Aristox. fr. 90 = Schol. Pl. Phd. 108d = DK 18.12; on the text, see above, ch. II 5,
n. 71. Also Euseb., Migne 24.746, Zenobius II 91, etc.; cf. O. Crusius, Philologus 52
(1893) 514ff. The ratio of the discs’ thickness was therefore 12:9: 8:6. Experiments
with discs are mentioned by Theo Sm. $7.7, Nicom. Ench. 6 p. 248.15, Por. In Ptol.
120.13ff, and (critically) Ptol. Harm. p. 17.18. According to Aristoxenus the musician
Glaucus of Rhegium played on the discs of Hippasus. There were also other comparable
cymbal-like instruments. A Diocles, who may be the Pythagorean of Phlius (Aristox.
᾿ - > f
. fr. 19), in the sth century discovered τὴν ἐν τοῖς ὀξυβάφοις ἁρμονίαν ἐν ὀστρακίνοις
ἀγγείοις (Suda s.v. Diokles; cf. Crusius, cited above).
34 Handbuch der Physik 8 (Berlin, 1927) 232; H. Gomperz, PhSt $7f; van der Waerden,
RE XXIV 279.
35 It is uncertain how much of what Eubulides reports about Hippasus goes back to
Hippasus himself (DK 18.14).
86 Theo Sm. $9.7ff. The source is neither Thrasyllus, who is copied in 47.18-49.5,
nor Adrastus, whose report (cf. 49.6, φησί in 50.4, 50.5-12 = Por. In Ptol. 7.24-8.5,
50.22~$1.4 = Por. In Ptol. 96.2-6) apparently extends at least as far as $7.10 (= Chalcid.
45) and possibly to $9.3, and is then taken up again at 61.17: ἐπανέλθωμεν δὲ ἐπὶ τὰ ὑπὸ
377
me ΚΕ LMAGOURDAN MUSICAL THEORY
(διάληψιν) of strings. ‘The experiment with vessels filled in varying
degrees is no more workable, with ordinary blows, than the one with
the hammers or the weights. ‘The cHvet is to be obtained by producing
a vibration in the column of air in the interior of the vessel, preferably
by resonance. Actually, the Aristotelian problema, in describing a
similar experiment only speaks of ἠχώ, echo;? and the resonance of
hollow vessels was used in the Greek theater.38 Thus it would be
possible that Lasus and Hippasus, in studying phenomena of resonance,
learned the numerical laws, and that a subsequent restatement in
somewhat cruder form transformed thcir action to one of striking the
vessels. Confidence is inspired by the fact that the next section discusses
experiments with strings.29
Lasus of Hermione, who became prominent in the time of the
Peisistratidae (Hdt. 7.6), was a close contemporary of Pythagoras. He
is never called a Pythagorean, but was doubtless among the earliest
Greck musicologists.4° What distinguished the Pythagoreans was
apparently not a special knowledge, inaccessible to others. Rather,
something which may well have lost its interest for professional
musicians came to be prized among them as a fundamental insight into
the nature of reality. The wondrous potency of music, which moves
the world and compels the spirit, captured in the net of number—this
was a cardinal clement of the secret of the universe revealed to the
wise Pythagoras.
τοῦ ᾿Αδράστου παραδεδομένα (cf. Schénberger 20 n. 1). The fact that the passage does
not come from the prevalent tradition of “Pythagorean” exegesis of the Timaeus
makes it all the more valuable.—ouvémeoOa ... τὰ τάχη... ἐν ἀριθμοῖς can scarcely be
right; συνέπεσθαι requires an object. Schmid (I 1.545 n. 2) conjectures συνέπεσθαι «τῷ
τῶν κινουμένων πάχει»; Schénberger argues against the assumption of a lacuna (26ff).
He is doubtless right in taking καὶ of περὶ τὸν Μεταποντῖνον Ἵππασον Πιυθαγορικὸν
ἄνδρα as an addition breaking the context: the sequel contains singular forms. One might
assume a rather long lacuna, in which a new subject had been introduced (Eudoxus,
according to Jan, Musici scriptores Sraect 131f; “Pythagoras” would be another possibility).
But in consideration of the context, Lasus can only have been cited for an experiment to
prove the numerical ratios. δίσκοι καὶ ἀγγεῖα are often mentioned together (Theo Sm.
$7.7, Ptol. Harm. p. 17.18, Por. In Ptol. 120.8fF ). F. Lasserre thinks experiments by Lasus
with the flute may be intended (Plutarque De la musique [Olten, 1954] 35ff).
” Arist. Pr. 19.50. ἀγγεῖον and ἠχώ go together for Aristotle (De an. 419b25). Bonitz,
Index aristotelicus s.v. ἠχώ, considers the word as equivalent to ἦχος in Pr. 19.50 and
19.42; but in 19.42 also the subject is surely resonance, in spite of the uncertainty of the
text.
ἣν ἠχεῖα. See Arist. Pr. 11.8-9; Heiberg 80; Tannery mentions them in connection
with the report on Lasus and Hippasus (MSc Ill 241ff).
3 sy.2tff; the experiment with weights is introduced later, with the words of δέ,
60.7).
“ Above, n. 12. Tannery, MSc ΠῚ 241f, drew the conclusion that the significance of
Pythagoras in the development of musical theory was less than is generally supposed.
378
ι. Speentation, Experimentation, and Miction
There iv no necessary connection between the discovery of the
musical ratios and the knowledge of the nature of sound, of vibration
or wave movement of the air, A nearly correct description is found in
the Sectio canonis of Euclid, the Aristotelian Problemata, and in the De
audibilibus:“' sound is a very rapid sequence of πληγαὶ ἀέρος, which are
disseminated as a result of the condensation and rarcfaction of the air;
the idea of the sound wave is attested as carly as the Stoa.*? A closer
succession of πληγαί makes a higher tone; and the concordsare correctly
explained by the simple coincidence of πληγαί, where the ratios are
simple. There was no attempt to measure the frequencies of vibration
experimentally.44
A much less fully developed theory is found in Plato and Aristotle.
Here too we find πληγαί and φορά, and rapid movement makes higher
pitch; but rapidity of propagation is confused with frequency, so that
higher tones are said to come to the hearer sooner than lower ones,
This conception is found in the Archytas fragments.4® Clearly there
41 See, in general, Schénberger 26-40 (E. Graf, Die Theorie der Akustik im griechischen
Altertum, Progr. Gumbinnen, 1894, is unsatisfactory). On the authorship of De audibilibus
(possibly Strato), see Zeller, II 2.95n.; Diring 1934, 169f; Wehrli, Straton 73£—Seet.
cat. proocm.: πάντες of φθόγγοι γίνονται πληγῆς τινος γινομένης, αἱ μὲν πυκνότεραι
(κινήσεις) ὀξυτέρους ποιοῦσι τοὺς φθόγγους, αἱ δὲ ἀραιότεραι βαθυτέρους. (Arist.)
Aud. 800a4: (τὸν ἀέρα) κινεῖσθαι... συστελλόμενον καὶ ἐκτεινόμενον.., 803b29: ἀεὶ
γὰρ ὁ ἕτερος ἀὴρ τὸν ἕτερον κινῶν... Pr. 11.6.899b3: ἡ μὲν γὰρ φωνὴ γίνεται ἡ συνεχὴν
ἀέρος ὠθουμένου ὑπ᾽ ἀέρος...
AGt. 4.19.4: ἐπειδὰν δὲ πληγῇ (ὁ ἀήρ), κυματοῦται κατὰ κύκλους ὀρθούς (the
comparison is with a wave in the water), καὶ αὕτη μὲν κυκλικῶς κινεῖται, ὁ δ' ἀὴρ
σφαιρικῶς.
4 (Arist.) Aud. 803b40: περισυγκαταλαμβάνεσθαι τοὺς ἑτέρους ἤχους ὑπὸ τῶν ἑτέρων
.., πλεονάκις... ἐν πάσαις ταῖς συμφωνίαις ὑπὸ τῶν ὀξυτέρων φθόγγων αἱ τοῦ ἀέρυν
γίνονται πληγαί, Pr. 19.39 (cf. 42), 921222, Nicom. ap. Boeth. Mus. 1.31. Cf van der
Waerden, Hermes 1943, 194ff.
** There is no way of verifying Capparelli’s idea (11 639) that the "Apyvrou πλαταγή
was a contrivance to measure vibration frequencies (DK 47Ar10). Could it have been a
kind of ratchet wheel, with a flexible pawl engaging a toothed wheel?
* evappovios φορά, Rep. 530d, ‘Tim. 47d. At ‘Tim. 67b we find the correlation between
faster movement and higher tone, as between slower movement and lower tone. At
Vim, 80a-b there is an attempt to explain why in spite of this we hear concords (the slower
tone finally catches up with the faster: καταλαμβάνει). Arist. Sens. 448aroff sets it forth
asa problem, λέγουσι... ὅτι οὐχ ἅμα μὲν ἀφικνοῦνται of ψόφοι. This is cautiously denied
De an. 42003 1ff, and more determinedly, from the point of view of the audibility of the
συμφωνίαι, Theophr. fr. 89 — Por. In Ptol. 63.19fF; cf also Arist. Pr. 11.6, 16, 20, 21,
62, 19.37, 30, 42.
“Archytas and Eudoxus ap. Theo Sm. 61.11 DK 47 Ὰ 10. =: 1964s Lasserre:
οὐ τὴν μὲν ταχεῖαν κίνησιν ὀξεῖαν εἶναι dre πλήττουσαν συνεχὲς καὶ ὠκύτερον
κεντοῦσαν τὸν ἀέρα, τὴν δὲ βραδεῖαν βαρεῖαν... Ptol. Harm. p. 20.0 says of Archytas:
μάλιστα τῶν 1]υθαγορείων ἐπιμεληθεὶς μουσικῆς. The calculation of the types of scale
made by Archytas (A16) and the proof from the properties of number transmitted by
Bocthius (Arg) may be regarded as authentic: but there are Brave doubts about tr. 1
(ch. also ch. ΠΕ αν an. 14, 20). Here Archytas ΠΝ praises his predecessors (like Hippoe,
379
“,. PYTHAGOREAN MUSICAL THEORY
was no thought of vibration numbers, for Archytas apparently assigned
the smaller number to the high tone and the larger to the low."
A certain Heraclides attributes the most advanced, and relatively
most correct theory of acoustics, the idea of tone as a multiplicity of
eee
Vict. 1.1, Hebd. 53); half of the fragment is taken up with their findings, in indirect
discourse. Thus Archytas appears in the role of a mere transmitter of Pythagorean wisdom,
as in the Ocellus fiction (D.L. 8.80). The word μαθήματα is used as a technical term, and
gocs beyond the usage of Plato; rot περὶ τὰ μαθήματα seem to have fulfilled the demands
of the Epinomis. It is odd that alongside the quadrivium—zepi yapetpias καὶ ἀριθμῶν Kai
σφαιρικᾶς καὶ οὐχ ἥκιστα περὶ μωσικῆς, there stands, independently, περί τε δὴ τᾶς τῶν
ἄστρων ταχυτᾶτος καὶ ἐπιτολᾶν καὶ δυσίων. What is this if it is not the content of σφαιρικάϑ
One sentence, DK | 432.7f, corresponds exactly to Pl. Rep. 5304, the single passage in
which Plato cites Pythagoreans by name. What a stroke of luck, if the source of this
particular passage has been preserved! But Plato is speaking here (and cf. Οὐαί. 4050)
only of the relation of astronomy and music (harmony of the spheres), whereas Archytas
obviously is intending to include all the μαθήματα. Especially suspicious is περὶ yap
ἀδελφεὰ τὰ τῶ ὄντος πρώτιστα δύο εἴδεα τὰν ἀναστροφὰν ἔχει (DK I 432.8f ). This appar-~
ently means πλῆθος (ποσόν) and μέγεθος (πηλίκον), exactly corresponding to the scheme
from which Nicomachus derives the quadrivium (Ar. 1.2f; δύο εἴδη, ibid. p. 4.20).
This kind of diaeresis of being is not even attested for the late Plato and Aristotle. Archytas,
however, speaks so briefly and allusively that one has to rely on Nicomachus as commen-
tary. On the other hand, not only does the enumeration of σημεῖα in the second part
make a good impression, but precisely the lack of clarity about the concept of speed
(rapidity of propagation or frequency of vibration) leads to the thought that Theophrastus’
polemic was directed against Archytas (fr. 89). Archytas combines the idea of “high”
tone with that of “strong” tone, which “can be heard further” (πόρσωθεν x’ ἀκούσαιμες,
DK I 434.13, cf. 433.15ff, 434.3f, but also Arist. Pr. 19.37, 920b25ff) against which Theo-
phrastus maintains that an equal amount, though a different kind, of force is necessary
for a lower tone (Por. In Ptol. 63.1ff, 63.20: εἰ, ὥς φασιν, καὶ πορρωτέρω ἀκούεται ὃ ὀξύτερος
φθόγγος τῷ πορρωτέρω διὰ τὴν τῆς κινήσεως ὀξύτητα διικνεῖσθαι). Still, Theophrastus
says of his opponents that they judged music not αἰσθήσει but τοῖς τῶν νοητῶν ἀριθμῶν
λόγοις (Por. In Ptol. 62.2£), while the Pythagoreans are too empiricist for Plato (Rep.
5310). Cf. Aristoxenus’ comment on his opponents’ musical theory: τὴν μὲν αἴσθησιν
ἐκκλίνοντες... νοητὰς δὲ κατασκευάζοντες αἰτίας, Harm. p. 32 Μ. This makes it seem that
Theophrastus is arguing against Pythagorean musical theory transformed from a Platonic
point of view, or on the basis of a Platonic treatment. In fact the vagueness of the Timaeus
on the concept of velocity exercised a continuing influence in the commentaries on the
dialogue; see Adrastus ap. Theo Sm. 50.5ff = Por. In Ptol. 7.24ff, and Ael. ap. Por. In
Ptol. 33.19ff.—In any case Frank’s statement is wrong, that Archytas “understood the
proportions of the intervals . . . as the relation of the vibration-numbers” (12, cf. 174ff »
“Αι Archytas A16 the greater number is associated with the lower tone. ΤῸ be sure,
Ptolemy always proceeds in such a way, in accordance with his canon experiments; he
could have converted the table of Archytas to correspond to his own system. In the
Vimacus there is no unambiguous indication how numbers and tones are connected (cf.
K ytzler, Hermes 87 [1959] 395ff); thus the problem of the authenticity of Philolaus A26
is not affected. But, since most experimental observations must lead to the association of
low tone and large number, this may—in line with the Archytas passage in Ptolemy—be
regarded as original (otherwise, Diiring 1934, 162 n. 2, who cites Sch6nberger; cf. the
next note.) On the other hand, Arist. fr. 47 (Plut. Mus. 1139c) assigns the number 12 to
the Nete, 6 to the Hypate (Pr. 19.23, 919br: διπλασία ἡ νήτη τῆς ὑπάτης), and Theophr.
fr, 80 says of the higher tone, πλείονας ἀριθμοὺς κεκινῆσθαι (Por. In Ptol. 62.146, 63.19£ ).
Similarly, Pr. 19.35, cf. also Plut. De an. procr. 1021e; in the contrary sense, Pr. 19.12,
23, 50; detailed treatment of the opposed principles of arrangement, Nicom. Ench. to,
Ρ. 254.
1, Speculation, Experimentation, and Fiction
immeasurably rapid atmospheric impulses, to Pythagoras himself;
since people identified hint with Hleraclides Ponticus,“" and since he
also cites Xenocrates, this attribution has found credence. As a conse-
quence, the opening sentence of the Sectio canonis becomes a quotation
of Pythagoras,” and the confusion of rapidity of propagation with
vibration frequency, in Archytas, Eudoxus, and Plato, becomes a
remarkable distortion of the long-known correct answer,® an error
into which all the leading minds of their age fell. However, the identifi-
cation of this Heraclides with Heraclides of Pontus is untenable.®!
Porphyry is copying out some rather late compendium, containing a
popular exposition of the way the inventor may have proceeded.*
Thus the whole is given as the reasoning of Pythagoras; Xenocrates is
cited at the beginning as a weighty authority, but scarcely more than
the one sentence can come from him.®3 This account has no more
48 “Ηρακλείδης ἐν τῇ μουσικῇ εἰσαγωγῇ, Por. In Ptol. 30.2-31.21, 32.23-33.4. Zeller
vacillated (II 1.1036 n. 1 for the identification, but I 509 n., against it), and Heinze (6 n. 2)
opposed the identification; but Jan, Musici scriptores graeci 135ff, Schénberger 118ff,
and Diiring 1934, 154ff argued in detail for it. Jan and Diiring see in Heraclides Ponticus
the discoverer of tonal vibration, Schénberger (113ff) and van der Waerden (Hermes
1943, 192) in Pythagoras. Cf. below, n. 51.
49 Heraclides: ἀνελθὼν ἐπὶ τὴν γένεσιν τῆς φωνῆς ἔφη ὡς “et (ὡσεὶ all texts; ὡς =
ὅτι = quotation mark) μέλλει τι ἐκ τῆς ἰσότητος σύμφωνον ἀκουσθήσεσθαι, κίνησιν
δεῖ τινα γενέσθαι." Sect. can. prooem.: .. εἰ ἄρα μέλλει τι ἀκούσεσθαι, πληγὴν
καὶ κίνησιν πρότερον δεῖ γενέσθαι. It is assumed by Diiring (1934, 155f) and van der
Waerden (Hermes 1943, 192) that Pythagoras is being cited here.
50 So expressly Schénberger 38f.
δ: Wehrli, Herakleides 113, gives the following arguments for a negative verdict: (Ὁ
Contradiction of Heraclides fr. 122. (2) Heraclides Ponticus can hardly be citing Xeno-
crates, who was more probably younger than he (already in Heinze); Heraclides did not
hesitate to attribute important material to Pythagoras, on his own responsibility (cf.
Burkert, Hermes 1960, 159ff). (3) “The detailed, systematic structure of the theory"
gocs even beyond Aristotle. We may add: (4) καταλαμβάνειν, “grasp,” “recognize”
(Por. In Ptol. 32.24), and ἀκατάληπτος (31.19), “unknowable,” are technical terins devised
by the Stoic Zeno (SVF I 60). (5) A title in the form εἰσαγωγή cannot be cited before
Chrysippus; and we may suppose that the pretentious Heraclides of Pontus did not busy
himself with schoolbooks. But this Heraclides’ εἰσαγωγή is pedantic, long-winded, and
characterless. Compare the passages (Arist.) Aud. 803b34ff: ai δὲ πληγαὶ γίνονται, . . πολλαὶ
καὶ κεχωρισμέναι, διὰ δὲ μικρότητα τοῦ μεταξὺ χρόνου τῆς ἀκοῆς οὐ δυναμένης
συναισθάνεσθαι τὰς διαλείψεις μία καὶ συνεχὴς ἡμῖν ἡ φωνὴ φαίνεται, and Heraclides
31.13: ἑκάστη τῶν χορδῶν πλείους προίεται φθόγγους (an awkward expression) ... ἀνὰ
μέσον τῶν κατὰ φθόγγους πληγῶν σιωπαὶ ἂν εἴησαν... ἡ δὲ ἀκοὴ τῶν μὲν σιγῶν οὐ
συναισθάνεται διὰ τὸ... τὰ διαστήματα μικρὰ ὄντα καὶ ἀκατάληπτα τυγχάνειν. The
laborious proof of the Sectio canonis that the vibrations must be related as whole numbers
(above, n. 4) is replaced here by the simple assertion (30.8) ἡ δὲ κίνησις οὐκ ἄνευ ἀριθμοῦ
γίνεται.
δ2..1..,.: καὶ οὕτως ἂν τις ἐπιδείξειεν, ,,
The φησί that keeps recurring in the Heraclides fragment is not referring τὸ Pythag-
oras (as Heinze 7), but to Heraclides; it is a sign that we have an epitome, and serves the
function of a new quotation mark, Ditring 1934, 1595f, thinks Didymus was the inter-
inediary source.
181
Fe ἃ δ Δ λ ἜΑ Σ “ΔΙῪ ψΔῳω ωΞ }, ITA VAY
historical value than the doxographical paragraph, in which Pythagoras
is put alongside Plato and Aristotle in the explanation of the nature of
govy,* or the report of Adrastus, who attributes to Pythagorcans the
Aristotelian theory of sound, using Aristotelian terms.
There were theories of sound and acoustics among the pre-Socratics
quite outside the Pythagorean ambit. A conncction of air movement,
resistance, and tone was obvious to anyone who considered the human
voice or a wind instrument. Alcmacon discovered the auditory canal
and the eardrum, and spoke of the “echo” in the interior of the ear,
though instead of air he spoke of the “void.”’5* Empedocles built onto
what Alcmaeon had achieved, speaking of the ‘“‘movement” and the
“ blows”’ of the air.5? This has been taken as an indication of the Pytha-
gorean origin of this doctrine ;** but scholars have overlooked the same
explanation of sound from movement of air and πλῆξις in Anaxagoras,5®
and no attention has been paid to a statement about Archelaus, the
pupil of Anaxagoras, who was active in Athens about 440 B.C.: πρῶτος
δὲ εἶπε φωνῆς γένεσιν τὴν τοῦ ἀέρος πλῆξιν.59 This report, about one of
the less prominent pre-Socratics, is important for that very reason. In
his casc, the doxography apparently found represented for the first
tinic a formulation that was later considered correct. It follows that
Archelaus must have gone beyond Anaxagoras and Empedocles, in a
way not made completely clear; but it can hardly mean anything more
than that he gave more exact information about the connection of
δά Aé&t. 4.20.1: Πυθαγόρας Πλάτων ᾿Αριστοτέλης ἀσώματον (τὴν φωνήν).
** Adrastus ap. Theo Sm. σοιόξ- Por. In Ptol. 7.228: ψόφος δὴ πλῆξις ἀέρος
κεκωλυμένου θρύπτεσθαι, cf. Arist. De an. 419b4ff, esp. broff, πληγή, 420a8 ὅταν δὲ κωλυθῇ
θρύπτεσθαι (ὁ ἀήρ), ἡ τούτου κίνησις ψόφος. What is worked out in detail in Aristotle
becomes a pithy definition for the ‘‘Pythagoreans.”
Alcmacon A5: ἀκούειν, τοῖς ὠσίν, διότι κενὸν ἐν αὐτοῖς ἐνυπάρχει" τοῦτο γὰρ
ἠχεῖν. Cf. AG.
Emp. A86 = Theophr. Sens. 9: τὴν δ᾽ ἀκοὴν ἀπὸ τῶν ἔσωθεν γίνεσθαι ψόφων"
ὅταν γὰρ ὁ ἀὴρ ὑπὸ τῆς φωνῆς κινηθῇ, ἠχεῖν evros... κινούμενον δὲ παίειν τὸν ἀέρα
πρὸς τὰ στερεὰ καὶ ποιεῖν ἦχον.
Schinberger 31f, van der Waerden, Hermes 1943, 192.
δ Anaxagoras A106: τὴν φωνὴν γίνεσθαι πνεύματος ἀντιπεσόντος στερεμνίῳ ἀέρι, τῇ
8° ὑποστροφῇ τῆς πλήξεως μέχρι τῶν ἀκοῶν προσενεχθέντος. Diogenes of Apollonia
also follows Empedocles, Α21: (τὴν ἀκοὴν γίνεσθαι) τοῦ ἐν τῇ κεφαλῇ ἀέρος ὑπὸ τῆς
φωνῆς τυπτομένου καὶ κινουμένου. According to Theophrastus (Sens. 59), the explanation
of φωνή as κίνησις τοῦ ἀέρος is the answer of “earlier” thinkers generally. He names
Hmpedocles, Anaxagoras, and Democritus. Democritus’ theory of sound was complicated
by atomistic ideas (A135), but in any case, τὴν φωνὴν εἶναι πυκνουμένου τοῦ ἀέρος καὶ
μετὰ βίας εἰσιόντος (ὃ- 5).
" DK 60A1 = D.L. 2.17.—Theo (59.9) ascribes to Lasus and Hippasus the explanation
of sound by “rapidity and slowness of movement” (above, n. 36); but this may be the
interpretation of the source, as the emphasis is on ἐν ἀριθμοῖς.
382
ι. Spectlation, Expertmentation, and Hletion
“blow” and tone, and accordingly the influence of the velocity and
strength of the “blow” on the pitch and intensity of the tone.
The acoustical theory found in Archytas, Eudoxus, and Plato is
therefore not a Pythagorean theory, but belongs in the general context
of Ionian φυσιολογία, Aristophanes, too, shows that speculation about
the origin of tones was in the air in the fifth century, when in the
Clouds he represents Socrates as investigating scientifically the buzzing
of a gnat and explaining it, ingeniously, by the principle of the
trumpct.®
If neither the recognition of the simplest numerical laws of music
nor physical theory on the nature of sound is exclusively Pythagorean,
then there only remains, as that which, in Plato’s eyes, distinguished
the Pythagoreans, the fundamental emphasis on number as such, which
led to development of the mathematical theory of music quite beyond
the requirements of actual practice.®® Later presentations of Pytha-
gorcan musical theory tried to derive as much as possible from a priori
considerations, and to refer as seldom as possible to experience and
experiment. Even the basic facts are—apparently—derived from
speculation, and everything else is derived from calculation of ratios.
The basic principle, at least from the time of the Sectio canonis and the
Aristotelian Problemata, was that musical intervals are expressed in the
form of “superparticular” or “multiple” proportions. The reasons
for the preferred position of the superparticular proportions are not
immediately obvious. It is based partly on the fact that all these
*) Ar. Nub. 156ff: κοῖλον πρὸς στενῷ προσκείμενον τὸν πρωκτὸν ἠχεῖν ὑπὸ βίας τοῦ
πνεύματος. This carries a verbal reminiscence of Democritus A 135, above, ἢ. 59:
στενόν-πυκνούμενον, Bia. Perhaps the ultimate source is Leucippus, from whom Diogenes
also borrowed (Theophr. Phys. op. fr. 2).
*’ Performing musicians work, according to Pl. Phlb. 56a, οὐ μέτρῳ ἀλλὰ μελέτης
στοχασμῷ. Arist. fr. 52 = Iam. Comm. math. sc. p. 80.158: of μὲν γὰρ τὰς ἀποδείξεις
καὶ τοὺς συλλογισμοὺς διωρισμένοι περὶ συμφωνίας καὶ τῶν ἄλλων τῶν τοιούτων ὥσπερ
οὗ κατὰ φιλοσοφίαν σκοπεῖν εἰώθασιν, οὐδενὸς δὲ κοινωνοῦσι τῶν ἔργων, ἀλλὰ κἂν τυγχά-
vwow αὐτῶν δυνάμενοί τι χειρουργεῖν, ὅταν μάθωσι τὰς ἀποδείξεις, ὥσπερ ἐπίτηδες εὐθὺς
αὐτὰ χεῖρον ποιοῦσιν.
"9 ἐπιμόριοι λόγοι are proportions of the type (n + 1): n; in Greek they are called
ἐπίτριτος, ἐπιτέταρτος, and so on. Like the λόγοι πολλαπλάσιοι (διπλάσιος, τριπλάσιος,
οἴς.), they are therefore “expressed in one word,” ἑνὶ ὀνόματι πρὸς ἀλλήλους λεγόμενοι,
Sect. can. prooem. p. 160.1ff, and this is regarded as adequate basis for the postulate.
Cf. Arist. Pr. 19.34, 41. The arbitrariness of the principle is obvious in the problem of the
cleventh (octave -| fourth = 8:3), which belongs among the concords but is not
recognized as such by the Pythagoreans. This was seen by Ptolemy (Harm. 1.6), who
therefore gave up the whole postulate. Actually, the important thing with the concords
is small whole numbers rather than ‘a single name.” See also Iam. Int Nic. 120.18ff,
Bocth. Mus. 2.18-20. §.8-9.
383
Ys PYETHAGOREAN MUSICAL THEORY
proportions were designated in Greek by a single word: but at the
sine time every ἐπιμόριος λύγος represents the connection of an odd
and an even number, and thus exemplifies the harmony of Limit and
Unlimited. So Pythagorean musical theory is intimately related to
numerical cosmology, and the importance of supcerparticular propor-
tion comes from its relation to number speculation in general.
The only empirical observations presupposed by the Sectio canonis
(10-12) for the derivation of the basic concords are that an octave
consists of a fourth and a fifth, and that, while a double octave is
consonant, a double fifth or double fourth is not. From these facts are
derived the mathematical proportions 2: 1, 3: 2, and 4:3. A proposi-
tion first proved by Archytas plays a central part in this,® so that the
whole procedure is credited to him, though Ptolemy, who briefly
recapitulates the proof, speaks of “the Pythagoreans” generally.%
In any case Archytas “devoted most attention, among the Pytha-
gorcans, to music’’ (Ptol. Harm. p. 30.9). Frank tried to show that
“there is not a trace, in Archytas, of the kind of a priori numerical
speculation” that was introduced into music theory by Plato (166).
In fact, only seven of the nine calculations of Archytas (A16) show
superparticular ratios; and in one point Winnington-Ingram has
shown an interesting connection between Archytas’ calculation and
musical practice.*? Tannery had argued that the postulate of “super-
particular or multiple proportions” was not early Pythagorean,®* and
Frank maintained that Archytas’ only concern was “to determine by
exact measurement the string lengths corresponding to the tones in
question,’’69
4 Above, n. 63; cf. ch. VI τ.
δ Arg; cf. Sect. can. 3; cf. ch. VI 2.
Ὁ Ptol. Harm. 1.5, p. 12.8-27, referred to Archytas by van der Waerden, Hermes
1943, 170; RE XXIV 279. Ptolemy advertises (pp. II.20-12.7) a “more speculative”
(λογικώτερον) derivation of the basic concords. The “best” ratio, 2: 1, corresponds to
the “most beautiful” concord, and the “ first two” superparticular proportions correspond
to the “first two” concords, the fourth and the fifth. Van der Waerden (Hermes 1943,
168, 198) extracts from this an axiomatic theory which he attributes to the Pythagoreans
before Archytas.
"7 CQ 26 (1932) 195~208; cf. During 1934, 251ff; van der Waerden, Hermes 1943,
185. The interval Parhypate-Hypate (28 : 27) of Archytas, with the Hyperhypate, which
is one whole tone lower (9 : 8), yields the pleasing interval of the diminished third (7: 6).
This very interval, Parhypate-Hyperhypate frequently occurs in the fragment of the
music for Euripides’ Orestes; so Archytas apparently took as his point of departure the
practical application of enharmonics.
"* MSc ΠῚ 76f. He stressed that it was not held to by either Archytas or Plato.
" Frank 266. At p. 157, Frank cites the ἀναμετροῦντες in Rep. $31a; but against this
van der Waerden correctly emphasizes (Hermes 1943, 176f) that in Plato’s view φθόγγοι
and συμφωνίαι are measured “against one another” (ἀλλήλοις), which sounds more like
the calculation of ratios than empirical measurement of the length of strings. One number
“measures” the other.
384
1, Speculation, Experimentation, and Fiction
It is hard to estimate, at this late date, how close Archytas was to
actual musical practice. ‘Phat he assigns the same pitch to the Parhypate
in all three genera corresponds to the uniformity of the name, and
comports with the method ofnotation.”° But it contradicts the evidence
of the senses, according to Ptolemy;7 and Aristoxenus is witness for
Ptolemy against Archytas in this point. Was Archytas depending more
on nomenclature than on the ear? In any case, proportions like
243 : 224, or even 32: 27 must have been derived by calculation rather
than exact measurement; and he must have based his calculations on
certain postulates. Even the fundamental idea that intervals are expres-
sible in proportions of whole numbers cannot be verified by measure-
ment. And quite exact measurement on the monochord (if Archytas
used this instrument at all; see above, n. 22) is required to decide
whether a double tone is 81: 64 or 80:64 = ς: 4.
Ptolemy says of Archytas that he begins with the postulate of super-
particular proportions, but then abandons his own principle.” Of
coursc it would be possible that Ptolemy ascribed to Archytas, wrongly,
the postulate which later came to be regarded as self-evident in
Pythagorean theory; but Archytas’ tenet that the superparticular
proportion is indivisible (A19) shows that this proportion played an
essential part in his music theory, for it is only important there. It
means that the octave, fifth, fourth, and whole tone cannot be divided
into equal parts, so that the arithmetic and harmonic mean take the
place of the geometric in music theory.”* And we have evidence that
the problem of means was important to Archytas.”4 Thus it is not a
coincidence that in Archytas’ table seven of nine proportions are
superparticular. How the exceptions came about remains a question.
Van der Waerden is right, then, in secing Archytas’ work as
During 1934, 253; Winnington-Ingrain, CQ 50 (19§6) 179; Vogel 1 84. Vogel
ascribes to Archytas’ calculations a basic significance in Greek music (47-57, 83-93).
But the oldest music theorists, who were only concerned with the enharmonic genus
(Aristox. Harm. p. 2 M.), must have been active before the day of Archytas, since he
takes all the 3 genera into consideration.
" Peo). Harm. p. 30.15 (cf. 32.3ff): ἀπάδων δὲ σαφῶς τῶν ἄντικρυς ἤδη ταῖς αἰσθήσεσιν
ὡμολογημένων. =
ΤΣ Ptol. Harm. p. 32.1ff (ch 30.13ff): παρὰ μὲν δὴ τὴν πρόθεσιν... αὐτῷ συνεστάθη
TO... χρωματικὸν τετράχορδον.
13. Dividing the octave according to the arithmetical and harmonic means gives the
famous series 6 8 y 12 (¢ a " ο΄; cf. Arist. fr. 47), the τελειοτάτη ἁρμονία which Tam.
In Nic, 118.19fF tells us was brought by Pythagoras from Babylon.
On the “harmonic mean” see Hippasus and Archytas, DK 18.15, Philolaus Aig,
and below, ch. VI 2. Archytas’ fr. 2 offers no impediment. For the application of the
neans in music, sce also Tim, 36a, Epin. ggoe, which points back to Archytas again. Por
the attempt to derive the table of Archytas (A16) from the doctrine of means, see van der
Waerden, Hermes 1943, 1841} (following ‘Tannery, MSc ΠῚ 104); Vogel 1 52-$3.
385
V. PYTHAGORBAN MUSICAL THEORY
belonging mainly to the development of the mathematical theory of
music.”> How successful he was in this, mathematically, may be dis-
cussed later.”® Archytas’ comparison of the συμφωνίαι as to the degree
of harmony or cuphony they show, ends up in mere arithmology.”?
In any case, Archytas was not conducting investigations in empirical
physics, involving cxact measurements; nor was he a practicing
musician, Rather, some observed facts and some speculative a priori
postulates are manipulated in a logical way, but do not coalesce into
a complete system. Plato’s criticism of “the Pythagoreans” also hits
Archytas. The numerical series he sets up is intended to correspond to
actual, audible music, but the solution is not satisfying in every respect.
The principles involved are still not fully applied, and, on the other
hand, agreement with musical practice is not complete. Plato’s com-
plaint does not mean that the Pythagoreans did not employ any kind
of speculation in their discussion of music, but that they hastily com-
bined λόγος and αἴσθησις, in their concern about the ἀκουόμεναι
συμφωνίαι, and therefore were forced to compromise in both fields.
2. NUMBER SYMBOLISM AND CALCULATION OF PROPORTIONS
IN PHILOLAUS
“Entaché d’absurdités mathématiques et d’erreurs pratiquement
énormes’’—this is the judgment of such an expert as Tannery? on the
musical system transmitted to us under the name of Philolaus. Never-
theless, Tannery, whose thesis was later developed more fully by
Frank,? relicd mainly on a different argument to prove the spuriousness
of the fragments in question. This was, that the calculation of a
diatonic scale to which Plato refers in the Timaeus was Plato’s own
work, not something borrowed from earlier Pythagorean calculations.
From this it followed, in his view, that since the Philolaus fragments,
ἴδ Van der Waerden, Hermes 1943, 166ff.
74 Of. below, ch. VI 2.
” Archytas At7 = Por. In Ptol. 107.15ff; the intermediate source is Didymus. Ptolemy
(Harm. 1.6, p. 14.1ff) argues against this method, without naming names. The idea of
ὅμοια and ἀνόμοια reminds of the ὁμοιοῦντες καὶ ἀνομοιοῦντες of Pl. Rep. 546b (the
“nuptial number”), where the concept of πυθμήν also occurs.
1 MSc Il 223.
4 Frank 263-277.
* Tannery concludes, from the contradiction between Plato and Archytas, that there
had not been in existence any earlier Pythagorean calculation, especially since Archytas
(fr. 1) says he is following—not correcting—his predecessors (above, ch. V 1, n. 46).
Frank develops the idea of the contradiction between Pythagorean “empiricism” and
Platonic “apriorism” (above, ch. V 1, n. 13).
386
,. Number Symbolism and Proportions in Philolaus
unlike Archytas, agree with the Timaeus, they must be a post-Platonic
forgery.‘
The seale of the ‘Timaeus, which has only equal whole-tone intervals
in the ratio ἡ: 8, and therefore pure fourths and fifths, but no thirds,®
is in Frank’s opinion ‘a purely speculative figment,” which has
“scarcely anything in common with the scales of real Greck music.’
Ptolemy is differently minded. He introduces the διάτονον διτονιαῖον
explicitly in relation to practicing musicians, who find it especially
“useful”; it is quite common in the tuning of the lyre.” Actually such
tuning is most easily accomplished by the use of fourths and fifths.®
Ptolemy adds that musicians, without regard to accurate theory, speak
of the “‘half-tone’”’ between Hypate and Parhypate;® and this corre-
sponds exactly to the conception of Aristoxenus, who certainly did not
take his description of the diatonic genus from the Timaeus. In fact
the word “diatonic,”’ διάτονος, means precisely that this scale is con-
structed διὰ τόνου, tone by tone; the alphabetic series of note designa-
tions, too, follows the diatonic series. The conception that the diatonic
tetrachord consists of two equal tones plus a “‘remainder”’ is therefore
the earliest, rooted in the practice of musical performers® Any
ὁ The following is a table, for ready comparison, of the various calculations of the
diatonic tetrachord:
Philolaus A26, fr. 6 Archytas A16 Tim. 36a-b
Mese (a)
> 9:8 9:8 9:8
Lichanos (g)
> 9:8 8:7 9:8
Parhypate (f)
> 246: 243 28 : 27 256: 243
Hypate (e)
δ The scale of the Timaeus passed, by way of Boethius, into the musical theory of the
Middle Ages. Archytas knows the major third 5: 4 in the enharmonic genus (but not
in the diatonic). This is called by Frank ‘‘the decisive step in acoustics, upon which the
entire physical analysis of the scale depends to the present day” (167). The arrangement
of our scale is 9: 8, 10:9, 16:15, and here (9: 8) Χ (10: 9) = (5: 4); but, since Greek
music does not use triads, the third scarcely plays any part in it.
6 Frank 13; cf. 181ff, 268.
7 Ptol. Harm. p. 39.14ff; cf. p. 43.10ff, p. 80.16ff.
®“Our piano tuners still follow this procedure,” says van der Waerden (Hermes
1943, 190), rightly referring to Aristox. Harm. p. 55 M.; also see Sect. can. 17.
* Ptol. Harm. p. 39.18f.
Van der Waerden, Hermes 1943, 190.3, refers also to Aristid. Quint. p. 22 M. This
passage was traced to Damon by R. Schifke, Aristides Quintilianus von der Musik (Berlin,
1937); but the attribution proves to be untenable (R. Wagner, Arch. f. Musikforsch. 4
[1939] 316fF).—-Ptol. Harm. p. 30.188 emphasizes that the διάτονον διτονιαῖον was not
significantly different from his own values for the diatonic genus (9: 8, 10:9, 16:15);
on the other hand, Frank maintains (166) that the ditone (9: 8) Χ (y: 8) gives “ἃ dis-
sonance quite intolerable to our ear.” It seems to me more likely that Ptolemy tested this
out than that Frank did; or are we to suppose that Ptolemy was tone-deaf?
387
Yo FPYTHAGOREAN MUSICAL THEORY
calculation taking its departure from here must necessarily lead to the
figures found in the Timaens, and only the search for “superparticular”
ratios, along perhaps with keener observation, could discover a
difference between the two “whole” tones, This in itself is enough to
destroy the basis for Frank's rejection of these fragments.
We can get further by consideration of the inconsistency in the
system of Archytas which Ptolemy mentions.” According to Archytas,
the pattern of the chromatic genus is,
Mese (a)
> ρθη
Lichanos (g flat)
= 243: 224
Parhypate (f) "3
> 28:27
Hypate (ec)
In the effort to explain these remarkable figures, scholars have
pointed out that the two lower intervals together make a whole tone,}2
or that the interval Lichanos—Paramese is a pure fourth.* But the
rationale offered by Ptolemy has not been much noticed: ““ Archytas
obtains the second tone in the chromatic genus (g flat) ... with the
help of the tone that occupies the same position in the diatonic genus
(z); for, he says, the second highest tone in the chromatic genus stands
in the ratio of 256: 243 to the corresponding tone in the diatonic
genus.”"!4 This explanation is so odd that we can not attribute it to an
intermediary source or to Ptolemy himself, even aside from the plain
φησὶ yap (Ἀρχύτας). Who could have got the idea, instead of using the
obvious relationships pointed out by modern scholars, of introducing
a calculation so complicated and based upon a different genus, the
diatonic? We have no alternative but to recognize the derivation
Ptolemy gives as that of Archytas. He found the highest interval in
the chromatic tetrachord not by harmonic division and not by refer-
chee to the natural concords, but by the extrinsic addition of two
previously known values, that of the diatonic whole tone (9: 8) and
the ratio 256: 243—the “remainder” when two whole tones are sub-
"AIG — Ptol. Hari. 1.13.
Diiring 1934, 255; Vogel T 52. (243 : 224) x (28: 27) = 9: 8. Still, the series 6 us
(πον third), 15: 14, 28:27 would be more “beautiful”; here the lower tones taken
together make a whole tone (15:14) Χ (28:27) =: το: y (Tannery, MSc ΠῚ 234f).
Van der Waerden, Hermes 1943, 184f,
ἰδ το]. Harm. p. 31.2ff.
388
4. Number Symbolism and Proportions in Philolaus
tracted from a fourth.!" ‘Thus Archytas is presupposing two things: a
music theory which builds its scale by the addition and subtraction of
intervals, and a calculation of the diatonic scale by the numerical
values found in the Timacus.
This music theory may be identified with the one ironically rejected
by Plato, which sought to identify the smallest interval, as a standard
of measurement; and we may conclude from Plato, as Frank does, that
this theory was better known than the Pythagorean. The Aristoxenian
conception of the tonal continuum is in any case primary; both the
language of professional musicians and the beginnings of musical
theory are couched in its terms. According to this approach, the
diatonic tetrachord has the structure tone, tone, semitone; the chro-
matic, tone and a half, semitone, semitone; and the enharmonic,
ditone, diesis, diesis.16
But beyond this, Archytas presupposes a calculation of the scale by
ratios—a diatonic tetrachord with the intervals 9 : 8, 9 : 8, and 2 46: 243.
Since his own picture of the diatonic tetrachord is different from this.”
it can hardly be his own invention; but, since musical theory based on
numerical ratio is a hallmark of Pythagoreanism, Archytas must have
borrowed the value 256: 243 from a Pythagorean predecessor. This
much we can infer from Archytas quite without reference to the
Philolaus problem; but since the ratio 256: 243 does appear among
the Philolaus testimonia, Archytas must be regarded as the most
important witness to the authenticity of the Philolaus fragments
dealing with music.
This is really the point where difficult problems begin. First of all,
the remarkable terminology of the directly quoted fragment (fr. 6)
(4:3): (9: 8): (9: 8) = 256: 243. This “remainder” (λείπων, Pl. Tim. 360) is
usually called λεῖμμα: Plut. Procr. 1017f, Adrastus ap. Theo Sm. 67.6 et saep., Cens.
10.7, Ptol. Harm. p. 23.2ff et saep., Por. In Ptol. 120ff, τ $4.22, Nicom. Exe. 2 p. 267.9,
εἴς.
Above, ch. V 1; Pl. Rep. 530f.
‘* Archytas’ figures are 9: 8, 8:7, and 28: 27, all superparticular ratios. If Archytas
is adopting older calculations, it seems that he may have arrived at his results by a process
of trial and error. Since, in the diatonic genus, the older calculation led to the awkward
256: 243, he took 8: 7 for one of the intervals instead of 9: 8, leaving the remainder of
28 : 27. In the enharmonic genus, he adopted the figures 80: 64 = §: 4, instead of (9: 8)
x (9:8) ~ 81:64; so that he had the remainder of 28:27 and, in between, 36: 35.
Inthe chromatic tetrachord the correction was not so easy; instead of 32: 27 he had to
choose 301 25 -- 6:§ (minor third) or 35: 30 =: 7:6 (diminished third).
Frank did not wholly miss this; on the ratio 256: 243 he writes (271), “. . . numbers
which play no role at all in Archytas; implicitly, they are included between the chromatic
Καὶ and the diatonic g.”” The contradiction of the two clauses apparently did not draw
Prank’s attention.
389
ἁρμονία, δὲ ὀξειᾶν, συλλαβί π᾿ place of the usual διὰ πασῶν, διὰ πέντε, διὰ
τεττίίρων; and δίεσις instead of λεῖμμα for the “smaller semitone”:
and above all Trite instead of Paramese for the tone which is separated
by a fourth from the Nete (c’), and by a whole tone from the Mese
(a)—namcely δ. Are these expressions, in this application, ancient and
genuine, or are they unfortunate, artificial archaisms?
Some are unquestionably carly technical terms of professional
musicians: ἁρμονία (a scale one octave long),?° συλλαβή, fourth (from
the manner of holding the lyre strings in the hand),?! and δι᾽ ὀξειῶν,
fifth (the play in the strings that brings up the tetrachord to an octave).2
In particular, συλλαβή and δι᾽ ὀξειῶν occur in an obviously Pythagorean
passage of the Hippocratic book De victu.28 Furthermore, the use of
these terms by the Pythagoreans is attested by Theophrastus: of μὲν
Πυθαγόρειοι τὴν μὲν διὰ τεσσάρων συμφωνίαν συλλαβὴν ἐκάλουν, τὴν δὲ
διὰ πέντε δι᾽ ὀξειᾶν, τὴν δὲ διὰ πασῶν τῷ συστήματι, ὡς καὶ Θεόφραστος
ἔφη, ἔθεντο dpuoviav."4 Then it is frequently mentioned in the Timaeus
19 δίεσις is mentioned repeatedly by Aristotle as a smallest interval (An. post. 84b39,
Met. 1o16b18ff, 1053a12, b35, etc.). In Aristoxenus a δίεσις is a third tone or quarter
tone (Harm. p. 46 M.); in any case it is smaller than a semitone.
*'On the following cf. Boeckh 65ff, Frank 273ff (though they fail to take account of
Hippoc. Vict. 1.8). On ἁρμονία see Arist. fr. 47 (the ἁρμονία consists of the numbers 6, 8,
9, 12), Pl. Phib. 17d, Aristox. 2 p. 36 M., Aristid. Quint. 2 p. 91 M.: διὰ πασῶν, ὃ καὶ
καλοῦμεν ἁρμονίαν. Similarly also Arist. Pr. 19.44, 47. Aristoxenus states (Harm. 2, p.
45 M.) that the ancient scales only extended for 1 octave. Van der Waerden, Hermes
1943, 176.2, correctly prefers the translation “scale” (rather than “ octave”).—Frank’s
assertion (268) that the appellation ἁρμονία for the octave presupposed Plato’s philosophy
of number, including the One and the Indefinite Dyad, is wrong; the word is old, and
the ratio 1:2 is a physical fact. (On ἁρμονία see also H. Koller, MH 16 [1959] 238-248).
*! Frank (273) finds an indication of late date in the term συλλαβή. There is presupposed,
he thinks, the ‘““Democritean” simile comparing tone and letter, interval and syllable,
σύστημα and word (cf. Frank 167ff; Burkert, Philologus 1959, 177.2). This is the way
Aclian explains the expression at Por. In Ptol. 96.30 (cf. Nicom. Ench. 9. p. 252.6f, Procl.
In Remp. I 231.1, Olympiod. In Phd. p. 169.16ff Norvin): but Aelian goes on, κατὰ
δὲ τοὺς ὀργανικοὺς λυρικοὺς συλλαβὴ εἴρηται ἀπὸ τοῦ λυρικοῦ σχήματος τῆς χειρός,
ἐπειδὴ ἐν τῇ ἑπταχόρδῳ χρήσει ἡ πρώτη σύλληψις τῶν δακτύλων κατὰ τὸ διὰ τεσσάρων
ἐγένετο σύμφωνον (Por. In Ptol. 97.286). This is to be understood quite concretely, and the
reference to the practical use of the lyre is to be preferred to the tortuous explanation
from the letter simile.
*2 The expression is found at Arist. Pr. 19.34, 41. cf. During 1934, 179: ‘‘doubtless
primarily the interval that filled out the original tetrachord . . . to an octave, progressing
from lower to higher.’ Thus one plays διὰ τεττάρων, δι᾽ ὀξειῶν, διὰ πασῶν.
*8 Vict. 1.8; cf. above, ch. IH 2, ἢ. 114.
*4 Acl. ap. Por. In Ptol. 96.21ff. One might ask whether Theophrastus mentioned
appovia alone, or also the two other terms; but, in any case, he was speaking of striking
Pythagorean language, and all 3 glosses are also cited in Nicom. Ench. 9, p. 252.5ff
(quoting Philolaus fr. 6) and Aristid. Quint. 1 p. 17 M.: παρὰ μέντοι τοῖς παλαιοῖς τὸ
μὲν διὰ τεσσάρων ἐκαλεῖτο συλλαβή, τὸ δὲ διὰ πέντε δι᾿ ὀξειῶν, τὸ δὲ διὰ πασῶν ἁρμονία,
Also see Hesych. s.v. δι᾽ ὀξειᾶν (from Diogenianus): παρὰ τοῖς Πυθαγορικοῖς λέγεται.
Notice the Doric form in Hesychius and Aelian. Hesychius s.v. πέραινον also seems to
refer to Philolaus (above, ch. III 2). Frank 276 concedes that Theophrastus is the terminus
ante quem for the fragment of Philolaus.
390
commentarios that the interval 256: 243, usually called λείμμα, had been
called Slew by the Pythagoreans.2> These are, then, ancient technical
terms of professional musicians whose use by Pythagoreans is carly
attested. Still, the question might be raised, whether Theophrastus
was thinking of Philolaus’ book, or some other work, in which case
some later forger made sophisticated use of the lexicographical glosses.
The situation of Philolaus’ Trite is more complicated, and indeed
leads to problems of the origin of the Greek musical system which are
mainly unsolved, and, given the present state of the source material,
insoluble. Boeckh saw here a point “so deeply recondite that it cannot
be attributed to any ordinary composer of pseudepigrapha’’ (70), while
Tannery thought, “Il est ἃ craindre que l’auteur n’ait trop archaisé”’ ,2
and Frank speaks of a “commonplace which can be found in almost
any popular writer on music of late antiquity” (275). The problem is
that of the seven-stringed lyre.
According to the ordinary system of Greek music, the basic tetra-
chord (e-a) is continued by another, either synemmenon (a-d’) or
diezeugmenon (b-e’). In one case the Trite is b flat, in the other c’,
but in Philolaus it is b, a whole tone from the Mese and a fourth away
from the Nete. It is no wonder that, according to Nicomachus, many
accused Philolaus of an error.2’ This very fact, however, makes it
improbable that some forger has injected an artificial archaism; the
purpose of an artificial patina is to arouse confidence, not mistrust.
So, before deciding that this is a mistake stemming from sheer stupid-
ity, we should try to interpret the name in a way that makes sense.
It is certain that the lyre, for a long time, had seven strings,?* and
that the number of strings was gradually increased in the fifth and
fourth centuries B.c., but next to nothing is known about how these
seven strings were tuned. From the time of the Aristotelian Problemata
the theory is attested that the seven-stringed lyre embodied the synem-
menon system, and that that of the diezeugmenon was introduced later;
Nicomachus attributes this step to Pythagoras.?® To this extent we can
28 Theo Sm. $4.11, 56.18, Chalcid. 45, Macrob. Somn. Sc. 2.1.23, Procl. In Tim. I
168.28, Boeth. Mus. 2.28.
26 Tannery, MSc ΠΠ 240. .
27 Nicom. Ench. p. 253.12: of δὲ τοῦτο (i.e., the explanation of Nicomachus, below,
n. 30) μὴ συνιέντες αἰτιῶνται ὡς οὐκ ὄντος δυνατοῦ ἐν ἐπιτρίτῳ λόγῳ εἶναι τρίτην ἀπὸ
νήτης.
28 Cf. above, ch. IV 4, ἡ. 6.
29 It is presupposed in the way the question is put at Ar. Pr. 19.7, 47 that the Nete was
missing in the ancient heptachord (on the text, see below, n. 36); cf. also section 25.
Nicom. Ench. 3 p. 241; 5 p. 244.14ff, cf. 7 p. 249.20ff, 11 p. 257.17ff, Boeth. Mus. 1.20.
The “ancient” heptachord synemmenon (e f g a Ὁ flat ο΄ d’) is clearly described in
391
accede τὸ Frank's use of the word ὁ ommonplace’; but this does not
explain Philolaus’ rite, In addition, the whole theory scems suspicious.
It is hardly credible that the most impressive of the musical intervals,
the octave, was not used in the early systems. The expression διὰ πασῶν
says in itself that “all the strings” together make an octave, and not a
seventh. An cight-stringed lyre is scarcely mentioned; it follows from
this that the seven-stringed lyre comprehended an octave, and this
means that one note of the scale was missing.
Nicomachus makes a twofold, and very tortuous, explanation of
Philolaus’ enigmatic Trite :3°
By Trite he means here the Paramese in the heptachord, before the
insertion of the dividing tone in the octachord. (1) This [i.e. the
Trite = Paramese} was separated from the Paranete (d) by an
undivided interval of one and a half tones.*! The inserted string took
away a whole tone from this, and the remaining semitone was left
in the diezeugmenon between the Paramese and the Trite. Under-
standably, then, the old Trite was separated by a fourth from the
Nete (b-e’), an interval which now is closed by the Paramese (b)
instead of the other . .. (2) Others say, quite persuasively, that the
inserted tone was not between Mese and Trite but between Trite
and Paranete, and that it was now called Trite, while the old Trite
became Paramese in the diezeugmenon.
In both the interpretations that Nicomachus gives, the scale pre-
supposed is obviously e f g a b d’ e’.82 The only difficulty rises from
Nicomachus, and the transition to the octachord (ef gabc’d’e’) is described in the words
Πυθαγόρας παρενέθηκεν ὄγδοόν τινα φθόγγον μεταξὺ μέσης καὶ παραμέσης, p. 244.22{:
μεταξὺ μέσης καὶ τρίτης, Ρ. 253.15; τὸν ὄγδοον φθόγγον... μεταξὺ μέσης καὶ τῆς ἀρχαίας
τρίτης παρενέθηκαν, p. 257.18f. It is not specifically mentioned that the whole upper
tetrachord is thus changed in pitch by one tone. The names of the notes remain the same,
except for the differentiation of Trite and Paramese; perhaps Nicomachus’ ideas come
more from calculation than from hearing. Another interpretation is proposed by J.
Chailley, “L’hexatonique grec d’aprés Nicomaque,” REG 69 (1956) 73-100. He postulates
an original defective heptachord e f g a ς΄ (Paramese = Trite) d’ ε΄ (pp. 73ff), and finds
this in Nicomachus (77ff). He thinks Nicomachus kept jumping back and forth between
a (never described) defective heptachord and the synemmenon heptachord which he
described in detail: “In order to preserve the octave and so that the Mese might not be
distant by a fourth from the two ends,” Pythagoras inserted the new note between Mese
and Paramese.
9° Nicom. Ench. 9 p. 253.4ff.
Ἵ The MSS have ἡμιτόνιον. Meibom emended to {τρι)ημιτόνιον. This expansion is
necessary, because it is essential to what follows (one tone is taken away and a half tone
remains over), and also to the epithet ἀσύνθετον, which is incomprehensible if applied
to a semitone (especially since only the diatonic is in question here). Vogel (II 771.)
rejects the correction, and ignores the succeeding context.
° Cf. Jan, Musici scriptores graeci 81 n.
392
‘Frere
a ΤΣ το ρον Seat δι ΠΡ ΡΥ ge ae ee. eee eyes a Δ ie a heal νὴ
Nicomachus’ dependence on the “insertion of the dividing tone” asa
transition from the synemmenon tothe diezeugmenon, which can contrib-
ute nothing but contusion, in this context.2 The clue to the correct
interpretation lies in recognizing the existence of an earlier, defective
scale with only seven notes but the range of an octave, in which one
note was missing in the upper tetrachord, the later Trite, ο΄. Now we
have, in fact, further traces of such a scale. The question is posed in the
Aristotelian Problemata, referring to the synemmenon: “Why did the
ancients, in their seven-stringed scales (ἁρμονίαι) retain the Hypate,
and not the Nete?’’** But the answer offers the consideration that
both notes, Hypate and Nete, were present, and that it was rather the
Trite which was omitted, taking us back to the scale e f g a b d’ ς΄ .35
The same interpretation is to be given the statement that Terpander
‘removed the Trite and added the Nete”’ ;36 and the σπονδειάζων τρόπος
of Olympus, in which the Trite is supposed to have been lacking,
belongs in this context.%”
The history of Greek scales is to a great extent still obscure, and in
33 The first explanation is very puzzling. Nicomachus, as the second explanation shows,
is thinking of an interpolation between Mese and Paramese (= Trite), which comports
with his own system (above, n. 29); but it is hard to see how he could say that the in-
sertion of an interval could ‘“‘take away”’ anything from the interval Paramese—Paranete.
94 (Arist.) Pr. 19.7; the names of the notes are given as in the σύστημα τέλειον. Nete
(= e’) is lacking in the synemmenon system.
35 Jan, Musici scriptores graeci 81 n., and Chailley (n. 29 above) 96f believe that the
synemmenon Trite (b flat) was named as lacking, so that the series was e f g ac d ς΄;
but since the Nete named surely belongs to the diezeugmenon system, we are dealing, in
the question as well as in the answer (ἀμφοτέρας yap κατέλιπον, τὴν δὲ τρίτην ἐξήρουν),
with diezeugmenon notes, 1.6., the Trite diezeugmenon (c’) is missing.
36 (Arist.) Pr. 19.32: ἑπτὰ ἦσαν ai χορδαὶ τὸ ἀρχαῖον, εἶτ᾽ ἐξελὼν τὴν τρίτην Τέρπανδρος
τὴν νήτην προσέθηκε. The synemmenon system is again taken as primary (τὸ ἀρχαῖον),
but the name Nete is taken from the diezeugmenon system, and therefore also the Trite;
once more, c’ is omitted, and the series is e f g a b α΄ ε΄. Plutarch (Mus. 28.1140f) also
speaks of the invention of the “Doric Nete” (ε by Terpander.—Another defective
system is presupposed at Pr. 19.47, in which were lacking “what is now called the
-Paramese and the whole-tone interval,” that is, b. Chailley (loc. cit. above, ἢ. 35) puts
this down as evidence for his scale e f g a c’ d’ e’, but apparently without reading the
following context: ἐχρῶντο δὲ μέσῃ τῇ ἐσχάτῃ τοῦ ἐπὶ τὸ ὀξὺ πυκνοῦ, i.e. next above the
Mese (a) came a πυκνόν, certainly not a τριημιτόνιον ἀσύνθετον as in Chailley 3 theory.
The initial sentence, at 922b5, ἢ οὐ τὴν ὑπάτην... ἀφήρουν, is obviously corrupt; it
could be corrected with Gevaert to οὐ τὴν vyirnv... (4.6. perhaps e f g flat a Ὁ flat ς
flat e’), or to od τὴν νήτην μόνην (i.e. the synemmenon system e f gab flat ς΄ d’), with
an.
: 37 Plut. Mus. 19, looking back to 11 (= Aristox. fr. 83). The reconstruction of the
σπονδειάζων τρόπος is especially complicated, since there seems to be a tone assumed as
standing between ς΄ and d’. Cf. Tannery, MSc III 299ff; R. Winnington-Ingram, The
Spondeian Scale,” CQ 22 (1928) 83-91 (he accepts e f* a Ὁ c*, as also for Philolaus, main-
taining that e f g a Ὁ d’ ε΄ is later reconstruction intended to explain Philolaus; p. 87).
See also Vogel II 9-38.
393
V. PYTHAGORRAN MUSICAL THEORY
particular we shall not go into the question of the relation of the
number of strings to the structure of the scale.3® But we may regard it
as certain that “defective” scales are very old, that the upper tetrachord
was in the carly days not completely developed, but that from a quite
early date the ἁρμονίαι had the range of an octave—8id. πασῶν.39 There
are significant traces of a scale in which the later Trite (c’) was not
included, and among them we may count the fragment of Philolaus.
For it is incredible that a forger drew the conclusion, from learned
comments about the “ missing” Trite, that the Paramese was originally
τρίτη, expecting his readers to draw the same conclusion; he would
gain nothing but criticism for all his trouble. These considerations,
along with the archaic terminology, allow us to regard Philolaus’
fragment 6 as one of the oldest pieces of evidence for Greek music.
We still have to consider the “mathematical absurdities” reported
by Boethius. This seems on the face of it a very late source; but since
Boethius is virtually translating Nicomachus, and since the latter quotes
directly Philolaus’ fragment 2 and a part of fragment 6, one must
suppose that the long word-for-word quotation, fragment 6, and the
details reported by Boethius come from the same line of tradition and
stand or fall together.4°
The question is that of the finer division of the tone. Philolaus
divides the whole tone into two unequal parts, diesis, 256: 243, and
apotome; the difference between the two is called komma. According
38 Tt is certain that the player on the lyre (kithara) did not have a special string for each
tone he could play. (The principal evidence js that of Pherecrates ap. Plut. Mus. 1141d;
cf. Diiring, Eranos 43 [1945] 176-197; Winnington-Ingram, Lustrum 3 [1958] 26f, cites
tions till the sth century B.c.) had the strings e a b d’ e’, and here, according to Gombosi
(42.8), is where Philolaus fits into the picture. The 7-stringed lyre had, according to
Gombosi (43) e gabe’ de’: or rather, including the Hyperhypate d egab ε ε΄
(Diiring, Eranos 1945, 192f). But see Winnington-Ingram, CQ 50 (1956) 169-186, and
Lustrum 3 (1058) 15. The epigram of Ion (fr. 6 Diehl, cf. above, n. 28) speaks of intervals
ofa fourth on the 7-stringed lyre.
” Ancient theory, from the time of Aristoxenus, denied that there had been any such
development; “the ancients,” they said, had known all the notes, and only ἐξα on
aesthetic grounds, from using some of them. (Plut. Mus. 19, harking back to 11 = Αγερόκ;
ΤῊ ἡ aa ov δι᾿ ἄγνοιαν ἀπείχοντο τῆς τρίτης... ἀλλὰ δῆλον ὅτι τὸ τοῦ ἤθους
4 Above, ch. III 2.
394
2. Number Symbolism and Proportions in Philolans
to this, the apotome would be 2187: 2048, and the komma 4441:
§24288—pure frivolity.! Philolaus’ treatment is different:
He establishes as the basis of tone the number which first makes the
cube of the first odd number and was highly honored among the
Pythagorcans [i.c., 27] .. . a number which is separated by a whole
tone from 24.4... From this, then, Philolaus makes two parts, one,
which is larger than half, and which he calls apotome, and another,
which is smaller than half, which he calls diesis: later it was called
“the smaller semitone.” The difference between these parts he calls
komma. And, first, he thinks that the diesis consists of 13 units, because
this is the difference between 256 and 243, and because this same
number, i.e. 13, is made up of a 9, a 3, and a 1. Of these the one
assumes its station as the point, the three that of the first odd-number
line, and the nine that of the first odd square number. Thus when he
has for these reasons established 13 as the diesis, the so-called semitone,
he concludes that the remainder of 27, which consists of [4 units, is
the apotome. But since the difference between 13 and 14 is I, he
decides that 1 is to be established as the komma. But he makes the
whole tone consist of 27 units because 27 is the difference between
216 and 243, which are distant from cach other by a whole tonc.
Philolaus finds a special meaning in the numbers 1, 3, 9, 13, and 27,
to the detriment of their mathematical sense. Intervals are actually
determined by numerical ratios, and here a single number is meaning-
less; but Philolaus is deciding on the inner meaning of single numbers,
He seems to have calculated a diatonic tetrachord correctly, working
from the ratios 4:3 for the fourth and 9:8 for the whole tone to
reach the numbers 192 (Mese), 216 (Lichanos), 243 (Parhypate), and
256 (Hypate);#4 but then, for him, the diesis (Parhypate—Hypate)
“1 The apotome is calculated at Timacus Locrus, p. 211.6 Thesleff, and called μεῖζον
ἁμιτόνιον, both apotome and komma in Procl. In Tim. Il 180ff (ὡς of παλαιοί φασιν,
p. 184.2) and Boeth. Mus. 2.30 p. 263f; 3.4 p. 279 cf. 2.11 Ρ. 267; 3.9 p. 281ff; 3.12
p. 286ff; 3.15 p. 295ff—Frank’s statement (272) is nonsense, “tthe author’s use of the
expression apotome for the irrational gives him away,” on the ground that according to
᾿ Eudemus ‘“Theatetus was the first to devise the term apotome and introduce it into
harmonics.” In his treatment of complicated irrational magnitudes, Theactetus used the
term apotome for magnitudes of the form +/a—\/b, and derived it from the harmonic
mean (sic; cf. ch. VI 2), and this has nothing to do with music. The musical apotome
has still less to do with irrationality, cither when correctly calculated or in the incorrect
verston of Philolaus.
Philolaus A26 — Boeth. Mus. 3.§ p. 276.15.
9929:24 - 9:8, a whole tone.
44 This calculation is found, e.g., at Theo Sin. ROIS 216: 192. 2492216 0- OER,
395
6 SE ΦΟΜΑΝΣΆΣΜΗΥ ἸΌΘΙΚ. ΑἹ, LHBORY
“consists of” the number 13, and the whole tone of 27, because these
are the differences between the numbers occurring in the respective
ratios; and the fact that a whole tone may also be expressed by the
ratio 27: 24 (= 9:8) seems to him a confirmation—in spite of the
totally different role played by 27 in this case. Philolaus is not thinking
of the mathematical uses of numbers, but they are to him entities in
themselves, and he combines them by addition and subtraction, where
the proper procedure is the multiplication and division of ratios. Of
course this leads him out of the realm of commonly understandable
numerical relationships. He seems to assume that, because 256: 243 is
a diesis and 243 : 216 a whole tone, the diesis also extends from 243 to
230 or from 220 to 216, and the apotome from 243 to 229 or 230 to 216,
as though 256: 243, 243: 230, and 229: 216 were the same ratio!
Tannery thought, quite understandably, that one could scarcely
ascribe such mathematical absurdities even to a composer of pseudepi-
grapha.* According to Frank (271) this kind of thing cannot have been
written by “any mathematician, and least of all by a Pythagorean;
their principal achievement was in the discovery of the mathematical
principles which Philolaus so disgracefully betrays.” All the same
Frank does not hestitate to attribute all this to Speusippus, so that Bie
cannot help wondering how the latter could gain admission to the
Academy so ἀγεωμέτρητος.
Boeckh’s expedient, to suggest that Boethius was responsible for
the errors,4¢ will not work; of all extant authors, Boethius has the
most detailed—and correct—exposition of the apotome and komma.
Nor can the error be removed by the correction of one or another
of the numbers:4? though this be madness, yet there is method jn it.
The basic flaw is that again and again difference takes the place of
45 66 x . é
" Tannery, MSc HI 223: “... qu’on peut ἃ peine attribuer méme ἃ un faussaire.”
Boeckh 79f. He refers to Proclus, who names Philolaus in the context of the scale
47 Diels’ sentence, DK I 405 n., “The apotome has 17% units, not 14”’ (based on a mis-
understanding of Boeckh, p. 79), gives an ‘“‘emendation” worthy of Philolaus. But
aside from that, fractions are not allowed in Greek calculation of proportions (Procl.
In Tim. Il 184.6 excuses himself for taking this liberty). ᾿
396
4. Number Symbolism and Proportions in Philolans
proportion; in place of the calculation of proportions, the idea of
addable and subtractable lines takes the center of attention. ‘This
impression is strengthened also by the separate report that Philolaus
used the expression ὑπεροχή with relation to all the intervals.
Similar numcrological games are to be found in the discussions of
the Timaeus by Plutarch and Adrastus.4® The number 13 is the essence
of the leimma,®° the number 27 “‘is” the tone;®! and here Plutarch
cites the Pythagoreans as his authority. There is doubtless a close
connection between this and the fragment of Philolaus, and the re-
liability of Boethius is confirmed: the only question is, whether the
commentators on the Timaeus had an authentic book by Philolaus or
whether the Philolaus book was written, pseudonymously, on the
basis of the Timaeus commentaries.®2
But Philolaus goes even further, as Bocthius elsewhere reports.®
After the diesis has been correctly defined as the measure by which a
fourth exceeds two whole tones, and the komma as the excess of the
whole tone over the sum of two dieses, comes the statement, “‘schisma
est dimidium commatis, diaschisma vero dimidium diescos.” The
48 Philolaus Α2ς = Por. In Ptol. 91.4ff.
49 Plut. (De an. procr. 1027e) knows, of course, that in music theory the important thing
is proportion, but he also thinks that the consideration of the individual numbers is a
θεωρία χάριν ἔχουσα οὐκ ἀφιλόσοφον. Thus he finds a deep meaning in any combination
of the numbers given in the Timaeus.
°° Plut. De an. procr. 12.1017f: τὰ δὲ τρισκαίδεκα λεῖμμα, καθάπερ Πλάτων, (οἱ
Πυθαγορικοὶ éxédovv).—Theo Sm. 69.3 (probably based on Adrastus, who certainly
is behind the section ending at 66.18 [φησίν, 66.12], and also is named in 73.16 ) calculates
the leimma at 486: 512, and comments, τινὲς δέ φασι μὴ ὀρθῶς εἰλῆφθαι τούτους τοὺς
ἀριθμούς" τὴν γὰρ ὑπεροχὴν... μὴ γίνεσθαι ιγ΄, ὅσα Πλάτων εἴρηκε δεῖν ἔχειν τὸ λεῖμμα.
οὐδὲν δὲ κωλύει καὶ ἐφ᾽ ἑτέρων ἀριθμῶν τὸν αὐτὸν εὑρίσκειν λόγον... οὐ γὰρ ἀριθμὸν
ὡρισμένον ἔλαβεν ὁ Πλάτων, ἀλλὰ λόγον ἀριθμοῦ. Here the error is recognized, but Theo
himself falls victim to it (though clumsiness of expression is partly to blaine): τὸ διάστημα
τὸ τῶν avs’ (256) πρὸς apy’ (243), τουτέστι τὰ ιγ΄. . . (69.12f). At 69.15 the semitone is
equated with the ratio 17:16, and correspondingly we read at 70.16f, ἐν μὲν ἀριθμῷ
οὐκ ἀεὶ εἰς ἴσα τέμνεται τὸ ἐπόγδοον διάστημα -5 though 18:17 were the same as
17: τό; then the ὑπεροχή 27 is divided into 14 and 13 (70.18), and this is supposed to
show that the tone 243 : 216 is not evenly divisible. So in Nicom. Exc. p. 260f, the serics
tone, leimma, tone is calculated with the numbers 216, 243, 256, and 288; it is pointed
out that 13 (256 — 243) is not the half of cither 32 (288 — 256) or of 27 (243 — 216),
so that the tone is not evenly divisible. Similarly, in Macrob. Sonim. Sc. 2.1.22, the tone
9: 8 is said to be indivisible because y cannot be bisected. All of this is not far behind
Philolaus in mathematical absurdity; bute deficiency of mathematical talent cannot
serve as a chronological criterion. (Cf. n. 47.)
*' Plut. De an. procr. 14.1018¢: of Πυθαγορικοὶ τὸν τόνον ἐν τούτῳ τῷ ἀριθμῷ (sc. 27)
τάττουσι. 101ya ols (with the numbers 13 and 27) τὰ μελῳδούμενα μετροῦσιν εὐσήμως
οἱ μαθηματικοὶ διαστήματα, τὸ μὲν δίεσιν, τὸ δὲ τόνον καλοῦντες.
The latter solution is adopted by ‘Tannery, MSc ΠῚ 2gof, citing the passages of
Plutarch.
δ Bocth, Mus. 3.8 p. 278.1| (DK 1 410.4ff).
WV. ΡΥἹΠΑΘΟΙΔΙΑΝ MUSICAL THEORY
shock of this is less in the hairsplitting procedure of dividing even the
komma once more than in the nonchalance with which this process of
bisection is introduced. If the komma can be bisected, surely a whole
tone may; but it is a basic tenet of Pythagorean musical theory that
neither the whole tone, the octave, nor in general any of the “super-
particular ratios” can be divided into two equal parts. To be sure
Philolaus realized that the diesis 256: 243 is smaller than a ee
but when the calculation of proportions is abandoned, bisection ee
in by the back door. In Tannery’s opinion, to allow this is to disqualif
oneself as a Pythagorean.® :
Once more, however, there is a significance underlying the error.
The apotome is necessary for the construction of the chromatic tetra-
chord; in this case the series semitone, semitone, tone and a half is
defined more precisely as diesis, apotome, diesis plus a whole tone;
and this formulation is obviously presupposed by Archytas.*¢ The
bisection of the diesis, however, has to do with the enharmonic genus
whose structure, rather than quarter-tone, quarter-tone, ditone become
diaschisma, diaschisma, ditone.®” This makes it seem that Philolaus dealt
with all three genera, taking the usual, “Aristoxenian” conception as
his point of departure. It is to some extent restated in terms of propor-
tion, and correspondingly corrected, but this is not carried through
and the numbets are treated as addable magnitudes. The idea of lines
and distances replaces the theory of proportion, just as number sym-
bolism smothers mathematics. :
The coincidences with commentaries on the Timaeus, which were
to be found in the faulty determination of the diesis (above, n. 50), do
not extend to the further subdivision of the diesis and komma. There
was no necessity, from the point of view of the Timaeus, to go into
the chromatic and enharmonic genera; but the Timaeus commentaries
never fai :
er fail to state that the whole tone cannot be bisected®&—not even -
by subterfuge. Thus we are precluded from deriving the Philolaus
δά For the impossibility of dividin
g evenly a superparticular ratio, and in particul
the be see Archytas Arg, Sect. can. 3, τό, 18, Ptol. Harm. 1.10 Nicaotee ᾿
. 267ff, . In Tim. i ff. ἘΠ
4 : 7 rocl. In Tim. Il 179.8ff, Gaudentius 13, Boeth. Mus. 3.1ff; above, ch. V 1,
ὅδ Tannery, MSc Ill 233. The naiveté of Leuci
: ty, M. : cippus 1s analogous, in declari
atoms are indivisible because of their smallness (DK II 97.26, ht is Sree aren
Above, n. 14. Proclus, In Ti
; . 14. ; int. It 188.21, says that the apoto
the diatonic genus. ἡ μι νὰ;
δ Tannery, MSc ΠῚ 224f.
58 (Ὁ ; γα υ) ε νι
Cf. above, nn. $0, $4. For the indivisibility of the leimma, see Sect. can. 18.
398
4. Number Symbolism and Proportlons iu Philolaus
testimonia from the ‘Vimacus tradition; and Philolaus stands quite
alone in the later tradition with his statements about the sehisma and
diaschisma. But Archytas was the first to give a proof of the basic
tenct of music theory which Philolaus violates. If we consider this in
connection with the fact that Archytas presupposes the calculation of
the diesis at 256: 243, and the structure of the chromatic tetrachord
in the manner of Philolaus, it follows that the material introduced by
Boethius must have belonged to Pythagorean musicology before
Archytas. The very inadequacy of the mathematics of Philolaus’
system, which takes no account of the accomplishments of Archytas,
becomes a proof of its authenticity. Thus the results of our examination
of the philosophical fragments and the astronomical system are
corroborated in a third area: there have been preserved, from a book
of the Pythagorean Philolaus, written toward the end of the fifth
century B.C., some authentic fragments, partly in the original wording
and partly as reported by others.
If these Philolaus testimonia are genuine, the conception of Pythag-
oreanism which Frank and Tannery take as their point of departure
will at any rate have to be corrected. If a Pythagorean of the fifth
century B.c. could fall into mathematical inconsistencies, and even
commit gross errors, then the nature of pre-Platonic, or rather, perhaps,
pre-Archytean Pythagoreanism cannot have lain in exact mathematics
or in natural science, but in the interpretation of the world with the
help of numbers thought of as symbols. In fact the practical significance
of Pythagorean musical theory is minimal. Only the basic facts can be
established by observation, and it was impossible to transpose the
values established by calculation back into audible tones. From this
point of view it is not surprising that Philolaus’ results for the apotome
and komma were mistaken. The attraction and the significance of this
theory lie not in the theory itself but in the orderly, rational pattern
that it reveals. Order and pattern, however, which the human spirit
craves, are to be found not only in the form of conceptual rigor and
neatly logical structure, but, at an carlier level, in richness of mutual
allusiveness and interconnection, where things fit together “sym-
bolically.” Thus the interrelation of number and music can be conceived,
earlier than any mathematically oriented natural science and quite
apart from it, as an aspect of the universal orderliness of the cosmos.
In the Chinese culture, where the recognition of basic musical relation-
ships is developed into an ingenious and intricately varied numerical
399
EEE OEE ——— = TT, BRBva Vr Ss FOSS eS ἂν τ -- '΄] we ey - 45
structure, mathemauical precision is purposely avoided.5" Not ἢ
Philolaus alone does it become clear that the iniporesiie thi ἐμὰ
Pythagorean musical theory was not the function of the pro
but the meaningful numbers. Van der Waerden draws ee the
tetractys, which has its roots in the ancient stratum of the acusmata
tradition.%! The “Fourness” which is the “harmony” in which th
Sirens sing, suggests the numbers 1, 2, 3, 4, which group ee :
into the fundamental concords 2: 1, 3: 2, and 4: 3, and thus oe
hend the orderliness not only of music but of the Ἢ νειν atin
of these four numbers is 10, the “perfect” number The ait: ne
the acusmata is independent of Philolaus,®! and eee back nae ᾿
to the oldest stratum of Pythagoreanism; and the idea of εὐ aye
the cosmos is also of great antiquity. According to a report of Eudem :
the Pythagoreans emphasized that the fourth, the fifth, and the re
are comprised in the number 9, because 2- 3 - 4-- 0; aad here, too αἰ
clearly number as such, not proportion, that is the ΚΡ ἄν cam ω
Not only Hippasus but Archytas as well classified the intervals b :
᾿ mee πον ἢ The earliest Pythagorean musical theory x 5
ounded on mathematic pave
9 aca rr ee aR
this situation is never letely aband oe ae
ι completely abandoned. On this basis, accordin
to our information, Hippasus made certain experiments, and ete
in his effort to express Pythagorean lore in the form of (nian φυσιολογί
made individual statements about the numerical structure of ei
music, showing a truly remarkable mixture of calculation and numeri ᾿
symbolism, in which its “sense” is more important than its accu 7
Only in Archytas does a real mathematical number theory aren
of this, and, in its application to music, a certain sort of natural scienc
—though this is nearly incapable of further development or progres
since its effort is to discern unalterable order in what ee a
°° Cf. below, ch. VI 4.
60 Hermes 1943, 178f.
ἐ On astral immortality, above, ch. IV 4.
62 Eudemus fr. 142 = DK 58Br8.
63 '
Hippasus DK 18.14 (but see ch. V τ, n. 35), Archytas A17 (ch. V 1, n. 77)
400
V I, PV EMAROTEGT ἐὙΠΙΙΙΕΙ ἃ λέν ἡ WFwrrer ὙΜΒΥΨΨΨΤΟῸ
Mathematics
τ. DID THE PYTHAGOREANS LAY THE FOUNDATIONS OF
GREEK MATHEMATICS?
As pre-Greck mathematics has been rediscovered in Egyptian papyri
and Babylonian clay tablets, a clearer light has been thrown on the
outstanding achievement of the Grecks in the development of pure
mathematics. The Babylonians, in particular, had made considerable
progress in the accumulation of detailed knowledge, in practical
calculation, and in the solution of even rather complicated problems
in arithmetic; beyond question, the Greeks had much to learn from
them. But it was always single problems they were concerned with,
> without any theoretical explanation
. « «ς . 2
making use of certain ~ recipes,
© cannot even be certain that the
or even an attempt at proof; “w
Babylonians formulated theorems in general terms.” Some of the
“recipes” or formulas are inexact,” but this did not matter as long as
they provided a practically useful approximation. Only with the
advent of Greek geometry do we find the demand for generalized
and stringent proof, for a deductive system based on axioms and
postulates.? This is the system presented to us in the Elements of Euclid,
ἃ model which until the nineteenth century seemed not to require any
essential improvement. All later achievements, including those of
the Indians and the Arabs,‘ build on the foundations laid by the
Greeks.
The very importance and influence of Euclid, however, makes the
study of early Greek mathematics more difficult. What is preserved
for us is, as so often happens, the final accomplishment, and by its
very greatness it obstructs our vision of the earlier development. It
ΠΣ Becker, Grdl. 22; cf. MD 11: ‘Sammlung von Rezepten”; Neugebauer, ExSc 48:
“Babylonian mathematics never transgressed the threshold of prescientific thought”;
146: ‘Greek mathematics of the Euuclidean style is a strictly Greek development.”
2 For example, the Babylonian formulas for the volume of a frustum of cone and
pyramid are wrong (though the former is still used today by woodsmen for calculating
the volume of a tree trunk): van der Waerden, SA 75f.
3 On this see the important paper of Kurt von Fritz, ABG 1955.
4 Neugebauer, ExSc 166ff.
401
POE ΚΑ, Κλ ΚΣ INUMBBR THEORY
virtually doomed its predecessors to extinction ;° pre-Euclidean mathe-
matics has to be reconstructed piecemeal from isolated fragments and
allusions. The most important are, first, the mathematical passages in
Plato and Aristotle,® and, sccondly, the few fragments that have
been preserved from the ἀριθμητικὴ ἱστορία and the γεωμετρικὴ ἱστορία
of Aristotle’s pupil Eudemus.? The “catalogue of geometers” in
Proclus’ commentary on Euclid is thought, with good reason, to be
derived in the main from Eudemus.® In the van of geometers who are
known for their specific accomplishments stands Hippocrates of
Chios, who was active about 430 B.C., in Athens.® His methods for the
quadrature of lunes, which Eudemus describes in detail, show a well-
developed style in setting forth a proof and a considerable stock of
geometrical theorems, some of a rather complicated kind2° As the
first to write Στοιχεῖα, “arguments set in line,” he had a clear idea of
the deductive nature of mathematics! A contemporary of Hippo-
δ The two small, extant astronomical writings of Autolycus of Pitane (ed. J. Mogenet,
Louvain 1950; see also Neugebauer, ExSc 225f) are somewhat earlier than Euclid.
°C. Mugler, Platon et la recherche mathématique de son époque (Paris, 1948), on which
see Cherniss, Rev. of Metaphysics 4 (1951) 395-425; R. 5. Brumbaugh, Plato’s Mathematical
Imagination (Bloomington, 19 54); A. Wedberg, Plato’s Philosophy of Mathematics (Stock-
holm, 1955; only treats purely philosophical problems); T. Heath, Mathematics in
Aristotle (Oxford, 1949).
"Frr. 133-142, along with the fragment preserved in Arabic, below, ch. VI 2,
n. 8.
8 Fr. 133 = Procl. In Encl. 64.16-68.4. The most Important indication of its derivation
from Eudemus is Proclus’ remark (68.4), οἱ μὲν οὖν τὰς ἱστορίας ἀναγράψαντες μέχρι
τούτου προάγουσι τὴν τῆς ἐπιστήμης ταύτης τελείωσιν (i.e., until the time of Plato and
Philippus of Opus); Eudemus wrote before Euclid.
making Hippocrates’ proofs seem more sophisticated than they were (as a modern
historian of mathematics unhesit
Greeks).
™ See Burkert, Philologus 1959, 193ff: below, n. 98.
402
1, Pythagoreans and the Foundations of Green Mathematics
‘ 1 ‘ ; ᾿ ehigew 1"
tes was ‘Theodorts of Cyrene, who discussed irrational ee ‘,
areas | ns and Archytas ὁ
ater came ‘Uheactetus of Athens! anc
and a generation later came ae
Sates 6 Still a tittle later was Eudoxus of Cnidus, poe ae
| 1 ᾿ a 3 μ a CS,
himself the most brilliant Greck mathematician before Arc vinnie
| 1011 : S ΠΙΡΌΓΟΙΙΣ
no less by his work on the theory of proportion and ne at
i om t 111
application of the method of exhaustion than by his cpoch-ma β
> ary oO
ee of the planets.15 Eudoxus was a younger contcmpor ᾿ :
Plato. This was a period of especially rich development in Gree
) its hi int 1 : ntur
mathematics, which reached its high point in the following century,
the time of Euclid, Archimedes, and ie nagar rae:
f Chios the special characte
As early as Hippocrates ὁ 7 ;
the geometry ὁ
evidence. But how did :
mathematics was fully in | , ᾿
Hippocrates come to be? The usual view, held almost a
in j Pythagoreans, if not in
ought among the Py a
that its origin is to be s he I :
Pythagoras himself. Of course, the unreliability of the later seers
inclinati ater
ae Pythagoras, and the well-known inclination to — ᾿
account; bu
s, have not been left out o
achievements to Pythagoras, ieee |
conscience were
the demands of the critica
scholars have felt that 3 ! aa
adequately satisfied by speaking of ὝΕΣ eager δον
Ι f mathematics, ᾿ Pythz an
ae ounts of the history o
oras.® In acc | 3 | ees
thematics” is treated as a self-contained unit between Thales anc
ma
; ; appears in
ve th. sc. p. 77.25. Theodorus ap}
e addition of Iam. Comm. ma On ἢ ἢ
12 a 43; ae (lam. VP p. 146.8). For confirmation, reference eae
aa ae iL ἭΝ bee according to which Theodorus taught the lp ean ere we
sagas ‘ expressly that he was a pupil of Protagoras, who “ veerec i ee
ΣΘΕΠΕΚΛΘΒΕῸΣ Ἢ e: cf. 161b, 162a, 1830). Even if we take the Pythagorean ca ᾿ ἊΝ
: ‘ ’ ᾿ ν [ re ‘
mathematics vet 1, n. 40), there remains the question whether Theodorus ἐν ae
sie ee any more to his Pythagoreanism than Dicon did his Olymy
mathematical pr
; r; RE XI 74). cat .
regs aes oo aa Fritz, ΚΕΝ A 1353-1363. The most Hage
18 See esp. Sachs, : in Arabic (below, ch. VI 2, 1. 82).
served in Arabic ’
: is a fragment of Eudemus pre ΜΆ
τ of hana ΑΝ ΠΕ τῆι was the first solution of the problem of doubling
is princip
-- Al4.
; Eudemus fr. 141 = DK 47 ἜΣ
gee: Heath, Math. I 322ff; van der Waerden, SA 179ff; ene Pi et 1
il V u. Abh. 38f: “It was the Pythagoreans whose ἡ ἣν Ἷ τ ee
16 er, Vortr. u. . ; ἜΝ > Greeks”; 4:
᾿ξ tical sciences first ensured these studies a place τοῖν τὼ " peer nT
ae cats ciences, the Pythagoreans devoted themselves especially .. . te 7 eae
6c es ᾿ ς “ i ἡ τ 11 Ww-
aes to them its first significant treatment”; and, with this, ε ΠῚ themiatics in his
nae he first impulse to the momentous development of ma : ᾿ oa Me
ee tas ; a rd says, CQ 1923, 5, “Pythagoras was the discoveres 9 ΩΝ a ae
εν ee anny ThPl 38: “it is. . . difficult to reject the oy ns aH 7
" tics ; Ἑ ᾿ : ΝΣ ΟΕ "ς: ‘a ᾿ hworeans
leita the true founder of mathematical science”; Hasse-Scholz ; Pees ἐμὴ ae
laid ie foundations of Greek τἀ ρυτῖς Ά; ie ae eae 2 ae aera
apie athemiatics ὦ ‘ 5
: > founder of a great school of m Bp πεν πεν Sou sa ate lie
sacar ΤΕΣ mae Krank AP 6t [1940] 48: “Mathematics as a pee came into ἢ
s pure theoria’’; : ae Ae . Puth: “ans.
oid which was the accomplishment mainly of the Pythagorean
’
403
EEE EE EEE IEE ἀρ Ψ Ια Ψ ἀμ 2 δὲ δ ἡ, δ Ἢ, ἢ Ζ 42). Δ FT
Plippocrates, ; Is dated, . US | icatl
ui τ lta at least by implication, in the first half of
To be sure, the general belief in “Ὁ Pythagorean mathematics” h
gone without criticism. ‘lannery more than once expressed hin self
skeptically on the matter,'? and the mathematicians Gee oa
subjected the “geometry of Pythagoras” to a criti ee 7 i
i critical examination!8
wh ich had the notable result of redating the discovery of incomm
bility to the end of the fifth century. Though this chron canes
revision was disputed, the attribution of this discovery to P th ae
himself, made by Proclus, has not since been serious] defend ie
Sachs’s book had as its objective to “ get rid of the en of P i ee
the mathematician” (p. VI); and she was able to show an ae
the special problem of the regular polyhedra, how the tradition of
Pythagoras treatment of it had been derived from the Ti
obscuring the contribution of Theaetetus2® This result has been widely
accepted, though not enough attention has been paid to ieee :
general implications, and in particular those resulting from anal a
of the catalogue of geometers.”” Somewhat later Heidel made a ἢ a
start on the whole question of the Pythagoreans and Greek i
matics and worked out in a more adequate way the devel moe
non-Pythagorean, Ionian mathematics. os
No other branch of history offers such temptations to conjectural
reconstruction as does the history of mathematics. In ae a
every detail has its fixed and unalterable place in a nexus of r ion.
so that it is often possible, on the basis of a brief and casual eta
reconstruct a complicated theory.° It is not surprising, then, that i.
gap in the history of mathematics which was ἜΡΡΕΙ, u bs rt
study of the evidence about Pythagoras has been filled ᾿ ᾿ a
succession of conjectural supplements. They have been based ie ἢ
on the analysis of Euclid’s Elements, regarded increasingly as a prin :
rather than an original work, and, secondly, on conclusions dr ἐκ
from the argumentation of the Eleatics, especially Zeno, in a
'? HScH 381ff, MSc Il 200f.
ik
Roaaes she Aen = Meds “Die Geometrie des Pythagoras,” Bibl. Math III 9
—$4; nt
(1909-1910) 97~155. eckungsgeschichte des Irrationalen,” Bibl. Math. ΠῚ τὸ
18 Above, ch. I 3, n. 116.
“0 The standard ex i
ample is the reconstructi
(above, ch. IV 2, Ὦ. 2). ἘΠΟΒΑΘΕ nS P lanetary system of Eudoxus
404
1, Pythagoreans and the Foundations of Greek Matnematics
scholars have found a specifically mathematical sense, ‘Tannery was a
pioneer in both these lines of research; and the most influential
reconstructions have been those of Becker®® and van der Waerden.®
Becker’s results, especially, have come to be looked upon as a fixed
point in the history of Pythagorean science.”
We have already mentioned the general problems involved in
this kind of reconstruction, in the history of science. The fact of a
logical inference can never determine precisely the time or place
of the inference, or the person who made it; no matter how exactly it
may be determined what propositions Hippocrates of Chios assumes as
proven, this will not increase the likelihood that Pythagoras or certain
Pythagoreans discovered and proved these propositions. This is the
point at which the critical analysis of historical evidence must supplant
logical inference.?> Another consideration cuts even deeper: a mathe-
matically impeccable reconstruction is valid only to the extent that it
can be known with certainty that mathematical logic was at work in
the original formulation. In dealing with the beginnings of mathe-
matics, when mathematical logic was first being developed, it cannot
be regarded as certain a priori that a mathematician respected the
presuppositions and drew the conclusions that mathematical logic
would have dictated. Of course logic has been inherent in human
thinking from time immemorial, and perhaps especially in Greek
thinking. But abstract mathematics, making use of proof, was an
invention of the Greeks, and not part of their original mental equip-
ment. The manner of its development cannot be determined by
logical inference, since this presupposes the decisive point. In the
history of science the old maxim is still valid, that we must not ascribe
to thinkers, especially those of early times, “either the principles of
their consequences or the consequences of their principles.”
21 For the argument based on the Eleatics, see above, ch. III 3. In his evaluation of
“Pythagorean arithmetic,” Tannery renounced his earlier skepticism when he became
aware of Archytas Arg (1905: MSc ΠῚ 244ff), and decided for “‘l’existence, dés le temps
d’Archytas, d’Eléments arithmétiques développés sous la forme que nous nommons
euclidienne”’ (249).
22 OSt 3; cf. below, VI 2, nn. 468.
23 MtAnn 1947-1949, cf. below, ch. VI 2, nn. 98ff.
24 Thus Reidemeister (16ff) pushes all the evidence to one side and finds the true
“‘pythagoraische (sic) Arithmetik” in Becker’s reconstruction.
25 Above, ch. IV 1, nn. 14-16.
26 “Ni les principes de leurs conséquences ni les conséquences de leurs principes,”’ as
formulated in the 18th century by Charles Batteux (Mondolfo, Inf. 306), and cited
repeatedly by Tannery.
405
What is the origin of the hirinly rooted conviction that Pythagorea-
nism was the source of Greek mathematics? This question is casy to
answer: it came from the educational tradition. Everyone comes upon
the name of Pythagoras for the first time in school mathematics: and
Pythagorean, begins his Arithmetic, which was much used as a school-
book, with Praise of the Master. Boethius’ Arithmetic, drawn largely
from Nicomachus, also names Pythagoras in its first line. Likewise,
Gerbert of Aurillac mentions the name of Pythagoras several times
in his geometry;2® and among the patrons of their ars geometriae
medieval Freemasons include Pythagoras.3° The Ars geometriae bearin
High Middle Ages, even presents an early version of the Arabic
numerals as an invention of the “Pythagorici,” and describes the
the Pythagorean tradition is so rich.*! Finally, the early modern period
derived the astronomy of Copernicus and Galileo from Pythagoras.32
The general belief in the Pythagorean origin of mathematics thus
stems from the Neoplatonic and neo-Pythagorean scholastic tradition
of late antiquity. In evaluating this it is worth bearing in mind that
according to an earnestly meant statement of Iamblichus, even the
27 Mart. Cap. 7.729, 8.803.
58 Isid. Et. 3.2.
2° Heath, Math. | 366.
9° M. Ghyka, Le nombre d’or, rites et rhythmes pythagoriciens dans le développement de la
civilisation occidentale ἢ (Paris, 193 1) 66.
numerals (1 to 9, but no zero). See K. Menninger, Zahlwort und Ziffer Π5 (Gottingen,
1958) 231ff and 1328 The problem of the Boethian geometry, and especially the mensa
Pythagorica, has not been completely solved; see M. Manitius, Lat. Lit. d. Mittelalters |
(Munich, 1011) 27f; Heiberg 49; Schanz—Hosius, Rom. Lit.-Gesch. IV 2 (Munich, 1920)
1538. The literature is cited by Capparelli (I 518), who may well be the only one to
believe that Pythagoras himself invented the Arabic numerals. See M, Folkerts,
“Boethius,” Geometrie II: Ein mathematisches Lehrbuch des Mittelalters (Wiesbaden,
1969).
32 Above, ch. IV 3, Ὦ, 2.
406
' , ; fics
δ, PP MAgereans atid ine rornaations of Greer Matnematt
problem of squaring the circle was solved by eo. "Ἔ
carlier tradition is much more sparse, but it does take us back to the
threshold of the Hellenistic period. It is repeatedly stated that tree
was imported from Egypt; | lerodotus speaks of the N ᾿ porriese ᾿ ᾿
continuing necessity to resurvey the land,™ and Aristotle o . : ' ‘i
(σχολή) of the Egyptian priests, which enabled — — ke
speculation (θεωρία) 35 Now, as early as Hecatacus of A he ig
ras is represented as bringing to Greece from Egypt, along w
4 4 \
f ara Kal τὰ
doctrine of metempsychosis, τὰ κατὰ eet Peay θεωρήμ
i τοὺς a is 338 and ch later Anticlides the historian
περὶ Tous ἀριθμούς 35 and not mu ae
of Alexander wrote that Pythagoras brought geometry to per ec on,
aks Ces Sian
after it had been invented by the Egyptian king Moeris. a
of Colophon regards Pythagoras as primarily the coryp acs ὁ
88 and Callimachus alludes, though in an
geometrical astronomy,?8 an chu .
ambiguous way, to Pythagoras’ discoveries in geometry.3® To be s
the notable development of Egyptian mathematics is Kev el
and it is remarkable that so remote a writer as Anticlides should be
concerned to testify to the mathematical achievements of Pythagoras.
iti he opinion of
Ὡ . Simpl. Categ. 192.16ff, citing a certain Sextus. Perhaps t
"ὦ δου. ia. ee (Alex. Aphrod. ap. Simpl. Se oe atone ) oe
f squaring the circle would be solved when a τετράγωνος ἀριθμός ecame ὲ i: tie
see (On this idea, Nicom. Ar. 2.17.7, Theo Sm. 38, Iam. In Nic. 61, 946, :
Bek Hero Geom, 2 p.
: 17, p. 787, Diod. 1.69, 81, Iam. VP 158, |
ae ἐπὴν ape πὰ τ δι δι by Theuth: Pl. Phdr. 274ς. Egyptian Bahr elie
anes a ” are mentioned by Democritus, fr. 299—an interesting, and indecc
f. serene attested in a fragment of dubious authenticity. Cf. Heath, Math.
am ;
I rarf. :
35 Arist. Met. 981b23.
| V 670f).
= FGrHist 264F25 (cf. Schwartz, RE | .
᾿ Bees FGrHist 140F1 (Miiller, FHG I 212 had mistakenly taken this clause
ith the recoding citation of Timaeus, FGrHist 566Ε17): καὶ γεωμετρίαν ἐπὶ πέρας
Sis ae Πυθαγόραν), Μοίριδος πρώτου εὑρόντος τὰς eo τῶν ἐπ ae τ τς
whe Ἢ g ive ivdpov. The whole sentence m
v : λείδης ἐν δευτέρῳ περὶ ᾿Αλεξάνδρου. > seni :
fe eee foe Diogenes Laertius had no other reason to cite him except for his
ΤΟ ;
mention of Pythagoras. .
88 Hermesianax fr. 2.85ff Diehl:
... Πυθαγόρην ἑλίκων κομψὰ γεωμετρίης
εὑράμενον καὶ κύκλον, ὅσον περιβάλλεται αἰθήρ,
ἢ ἐνὶ σφαίρῃ πάντ᾽ ἀποπλασσάμενον. | !
(S ey ἠέ τονε! μὲ MSS, ἀπομασσάμενον Hemsterhuis, Diehl, ἀπομαξάμενον
So
Kaibel.)
39 Below, n. 106. a ee
“0 Modern scholarship rates the Babylonian influence on Greek mathematic
f), On
higher (see Heath, Math. I 122ff; Becker, MD 9; van der nearer ἼΩΝ Lied
ἐς other hand, the Babylonian tradition is scarcely mentioned in the ᾿ ἐξ Τὸ δυοῖν
aside from lan In Nic. 118.23; below, ch. VI 2, n. 89). But (Pe sae “ἢ ΡΒ)
ν 4 4 ; ig from the Phoenicians (Por. » : :
learned τὰ περὶ ἀριθμούς τε Kai λογισμούς
407
Ὧν «τυ σον ΠΡ» Ὁ Φυβιί ΒΑ Ὺ ἐψγϊ ἀμ Πς ἢ FIBUIK Y
Neither Herodotus
odotus Nor Lsocrates sees ; “CA8]
Ws sees ally Occasion to mention Pythag-
Oras In connection with Egyptian geometry, though they do 50 ἃ
speaking of Egyptian burial rites and aytoretar.4! Was there fic
ee m the ee of Pythagoras between Isocrates and dee
oF Mecatacus of Abdera and Anticlides—that is. j
Old Academy? The way the tradition about ae eo aire ἡ
be seen in ἃ sentence of Aétius, appended to his report of Hi m0 is
theory of vision: “Some also credit Pythagoras with a me in hi
doctrine, as being the chief authority in mathematics (βεβαιωτὴ :
μαθημάτων) .᾽᾽2 Here tradition is not being transmitted, but ee
85
Modern scholarship bases its judgment about
mathematician on certain other pieces of evidence t
more reliable. Tannery's point of departure was the
of when Pythagorean geometry” was committed to writing. |
IJamblichus we read, ἐκαλεῖτο δὲ ἡ γεωμετρία πρὸς ]]υθαγόρου “δι ᾿
and Tannery translated this, “geometry was called ‘the ie
according to Pythagoras.’”’4 In consideration of the cont ΠΝ
lamblichus, he interpreted this to mean that before Hippo a of
Chios there was published a treatise on geometry with the aa
Tradition according to Pythagoras. This Pythagorean textbook wa he
cornerstone of Tannery’s reconstruction, and it has continued t
a part right down to the present day.* It owes its existence. ho ale
3 an obvious mistake in translation; it is impossible to fee am
as ἱστορία together, and the meaning must be “ geometry ies call 4
ἱστορία by Pythagoras.” Thus the topic under discussion is Pyth i
use of words, and not anything about a book. Halk long ete!
the surprising explanation of this report. Heraclitus wrote
oras son of Mnesarchus practiced inquiry ( }
a 129). This sentence is given by
Pythagoras left writings, and thus it
Pythagoras the
hat appear to be
crucial question
“ Pythag-
? ἰστορίην) most of all men”
Diogenes Laertius as proof that
has been a link in the Pythag-
*' Cf. above, ch. II 3.
Δ Aét. 4.13.10 (cf. above, ch. I 2, πῃ. 76).
43 Tam. VP 89 = i
oe Bad . πως math, sc. p. 78. 5; on the connection between the two parallel
7 » ch. Il 5. The sentence cited here is obviously an 1 lation i
nistotelian context; cf. below ch. VI 3, n. 59 aa
| Tradition touchant Pythagore,” Tannery, Géom. 8rf: ‘
suivant Pythagore,” HScH 124; MSc VII τιοῦ |
45 Re . 5
y 227ff; Michel 81, 174f; van der Waerden, SA 116f; Szab6, Maia τὸ (1958)
[1 3 με
une geometrie, la Tradition
408
ι, Pythagoreans and the Foundations of Greek Mathematics
orean tradition. It was only natural for a later Pythagorean to draw
the conclusion: Pythagoras was neither a historian nor a geographer,
but βεβαιωτὴς τῶν μαθημάτων; therefore Heraclitus must mean
γεωμετρία. Thus the whole sentence is an erroncous philological
inference from a sentence of Heraclitus.** Even if this explanation were
not allowed as more than a possibility, there remains no firm basis for
the belief that Pythagoras was a geometer, and in any case no attestation
of his having written anything.
The chief testimony for Pythagoras as a mathematician, always cited
in the literature,*’ is in the “catalogue of geometers”’ given by Proclus,
whose principal source is rightly thought to be Eudemus.** “ Pythag-
oras turned its (geometry’s) philosophy into a form of liberal edu-
cation, seeking its first principles (ἀρχάς) from a higher source (ἀνωθεν)
and hunting out its laws by a nonmaterialistic and intellectual procedure
(ἀύλως καὶ νοερῶς) ...” The weight of this pronouncement is enhanced
by the prestige of Eudemus as a pupil of Aristotle, as well as by the
undeniable fact that the special character of Greek mathematics consists
precisely in its theoretical structure, as distinguished from the oriental
“recipes.” To be sure, the passage that follows, ascribing to Pythagoras
the discovery of irrationality and of the “cosmic bodies,” is less often
accepted ;4® but even in the sentence quoted there are suspicious
features. Does the phrase dvAws καὶ νοερῶς seem more like the phrase
of an early Peripatetic, or like a favorite theme of all Neoplatonists,
and especially Proclus?5° And does Aristotle not say expressly, of the
Pythagoreans, “‘they apply their propositions to bodies’’—bringing
out the distinction, in this regard, between them and all genuine
46 Lglk 8f. Thus it is not so that ‘‘it is utterly inconceivable how . . . anyone could
have thought of inventing anything of the sort” (von Fritz, SBM 1960, 20).—Ps.-
Hippoc. Ep. 22 uses the phrases ἱστορίη yewperpixy (1) and ἡ τῆς yewperpins ἱστορίη
(2), along with the word ἀρίθμησις; thus he knows the connection of yewperpin and
ἱστορίή, and uses it as an element of Ionic coloring.—Von Fritz (SBM# 1960, 20) interprets
the word ἱστορίη as reflecting the fact that Pythagoras collected specific pieces of mathe-
matical knowledge from oriental sources; and Frenkian’s interpretation is similar.
47 Eo. Heath, Aristarchus 46, Math. 1141; Rey, 216; Michel passim, esp. 168ff; Morrison,
CQ 1956, 153; Becker, Grdl. 22, MD 12. A cautious attitude is expressed by van der
Waerden (SA οοἵ; but cf. 100), by Wehrli (Eudemos 114; but at 115 the sentence cited is
interpreted as deriving from Eudemus), by Reidemeister 18ff, Heidel (AJP 1940, 16ff),
-and von Fritz (RE XXIV 198—but no-mention is made that the sentence comes from
Iamblichus). Timpanaro Cardini (I 30 n.) cites Sachs (whom Frank follows, 363 n. 209),
but without reason given concludes, ‘‘ritengo che sia da sostenere la derivazione da
Eudemo”; Iam. Comm. math. sc. is ignored.
48 Procl. In Eucl. 65.16 = Eudemus fr. 133 (cf. above, n. 8) = DK 14.6a.
49 Above, nn. 18-19.
50 ἄυνλος is attested once by Aristotle, voepds twice, but neither as adverb. Cf. Procl.
In Encl. 9.13, 63.5, 137-21, et saep.
409
ee ὮΝ eS ΜΙΝ
PI. “ope 25] νὰ :
tonists?™! And does not Ludemus, as far as we know from other
ὃ
fragments, always speak of Pythagore
De communi mathematica
times by the page in his
red with “Pythagorean
lie before Pythagoras, in the work of Egyptians, Assyrians, and Chald-
acans. The distinctive as |
pect of Pythagoras’ work is not only in new
subtlety, and exactitude "δ
ne a |
a Pega it 1s necessary to reconstruct, with considerable effort
w . ., :
at they would probably have said if one of them could have
ichus has no doubts ab
ἐν : ἣν ts about his
oplatonic theme: “If we are to pursue mathematics in the Pythago-
rean
— we must follow its upward path, full of divinity which
; 3
tings purification and perfection.” ὅτι τοίνυν οὐδὲ εἰκῇ Πυθαγόρας τὺ
2 THY
3 A) 4
πέρι Ta pad f ᾿ 7)
ρ μαθήματα φιλοσοφίαν εἰς σχῆμα παιδείας ἐλευθερίου μετέστησε
bd
51 Ae;
᾿ Arist. Met. 1083b18: on the other hand, Plato R
or the “Pythagorean” saying σχᾶμα καὶ βᾶμα, see
text above, and cf. above, ch. 1 2. Plato’s direct criticis
considered; above, ch. 1.
52 Ἑ
ats ae ss 60, 88, 136, I 37, 142, 146. For Aristotle, ch. I 2.
ee ara in N. Festa’s edition of Comm, math. sc. (Teubner 1891)
m. math. sc. Ὁ. 66.9£: ἐπεὶ δὲ τῇ f io
Paani τῆς Πυθαγορείου μὲν μαθηματικῆς προηγουμένως
ὅδ Comm. math. sc
, - SC. p. 67.6f (cf. the table of : τὴ
παρ εις © Of contents 6.21f): τὴν, ,
6 Comm. math. sc.
ἀκριβείᾳ. . ,
57
Comm. math. sc. p. 68.11ff: ,. . δεῖ τοιόνδε τι ποιεῖν-
ὁρμωμένους σωματοποιεῖν ἀεὶ τὰ τ
προσηκούσας καὶ τὰ παραλειπό Ἵ ῦ
po τὰ 1 μενα ἀναπληροῦν, στοχάζεσθαί L τὸ VY τῇ
gress γνώμης, τίνα ἂν εἶπον, εἰ ἐνεχώρει τινὰ αὐτῶν Fev nile ee
wee math. SC. 23, p. 70.1ff; ἐλευθέριος παιδεία δ
(ρ. τ 5 ) προῆγεν and ἀκρίβεια to the Preceding (p. 67.3fF )
at lamblichus just happened to quote Eudemus in the t
δ 7 ? “~ ~
. WOLOTHTA αὐτοῦ τῆς
- 67.226: καὶ μὴν ἀ
Ρ. 6.22: καὶ μὴν ἀποδείξεών γε καθαρότητι λεπτότητί τε καὶ
3 ᾿ ~
a70 σμικρῶν αἰθυγμάτων
3 ,ὔὕ 3 4
ApXas τε αὐτὰ ἀνάγειν τὰς
410
ι, Pythagoreans and the Foundations of Greek’ Mathematics
for practical life and in itself! Uhis may be based in parton Aristotle;
but, ifanything in it is original with lamblichus, it is the chapter division
and the transitional formulac."° In the succeeding passape there is
nothing about Pythagoras and the Pythagoreans except lamblichus’
introduction and his concluding sentence, “It was natural, then, that
for all these reasons the Pythagorcans honored the study of mathe-
matics. 6!
Therefore the often cited sentence about Pythagoras in the “catalogue
of geometers”’ is not from Eudemus, but is a formulation of Iamblichus,
as had been recognized over sixty years ago.®? Thus its “authority”
is precisely reversed; if, in a context whose significant parts arc obvious-
ly derived from Eudemus, the passage dealing specifically with Pythag-
oras has been supplemented with material from Iamblichus, this is
an indication that there had been a gap to fill, and that Eudemus did
not give enough information about Pythagoras or even nonce at all.
As far as concerns the specific discoveries attributed to Pythagoras, the
“cosmic bodies”’ are regarded by Proclus as the apex, or the quintes-
sence, of all geometry; and the discovery of irrationality had, long
before Proclus, been interpreted in a Platonic sense and bound up with
the thrilling story about mathematical secrecy and its betrayal and
the ensuing divine punishment.* Proclus has merely attributed to
°° Merlan, PlNeop! 141ff, referred Comm. math. sc. 23 to Aristotle's Protrepticus. ‘Ulin
was accepted by Festugiére, Rev. philos. 81 (1956) 117-127, but rejected by 1, Diring,
Aristotle’s Protrepticus (Goteborg, 1961) 209.
Ὁ Deubner showed that the division into chapters and the pretixed suniunaries were
the work of Iamblichus himself (SBBln 1935, 689f). Merlan (PINeopl 127) is inclined to
attribute the mention of Pythagoras to Aristotle, though he adinits (126) the likelihood
of some reworking by Iamblichus, precisely at the beginning and end; Festugidre (126;
see preceding note) also expresses doubts. In the parallel passage to which Merlin relers,
however (Arist. Protr. fr. 11 ἍΜ. == Iam. Protr. p. 41.781), the mention of Pythagotay is
another addition by Iamblichus (cf. Burkert, Hermes 1960, 1661).
* Comm. math, sc. p. 73.17; the passage between (pp. 70.7—73.17) is mostly put in the
~ first plural.
8 Vogt, Bibl. math. 1908-1909, 31f; Sachs 30ff (1917). A refutation of this hay never
been attempted and could scarcely succeed; the general denial by Friedlacnder (Platon Ι
[Berlin, 1928] 108 n. 3) was omitted from the second edition (I? [1044} 211. αν 15; Eng.
ed. 353 n. 15).
°° Procl. In Eucl. 70.24: περὶ τῶν κοσμικῶν σχημάτων ἐστὶν ὁ σύμπας τῷ γεωμέτρῃ
λόγος... τελευτῶν... εἰς τὴν ποικιλίαν τῆς τούτων συστάσεως (Cf. ὁς..0: τὴν τῶν
κοσμικῶν σχημάτων avoraow),—The terminology is therefore that of Proclus. Naturally
the latter was affected in this judgment by the Timaens, which for hin shared with the
Chaldean Oracles the honor of being the most important book in the world (Marinus
ν΄. νοεῖ. 38).
"Cf. below, ch. VI 3. There is no occasion to reject the reading ἀλόγων in fiver of
the weakly attested ἀναλόγων, as von Fritz does (Medan to4gs, 246 no 14). (Fiicdlein
refers not to any miamuseripe butte “alii,” named by EE. August in his edition of Euclid
[Berlin, 1826].)
411
Vie PYTHAGOREAN NUMBER THEORY
vel): as the
Pythagoras the two most famous and, from the Platonic point of view
ΝΜ ‘a : “, ‘wer ; ᾿ ;
most significant achievements of geometry, supplying an illustration to
accompany the generally phrased sentence from Iamblichus. Nothing
is left of the supposed testimony of Eudemus to the achievement of
Pythagoras in the foundation of mathematics.
| Another testimony, supposedly coming from Aristotle himself, is
oe scarcely Ye often, as evidence for Pythagoras as a mathemadcan
t is given as follows in the octi |
“After these men, phe eth ame peti ae eae
: nesarchus
first worked at mathematics and numbers, but at some later joe
he also indulged in miracle-mongering like that of seen a8
This comes from the Historiae mirabiles of Apollonius, who has ἘΠ
finished dealing, successively, with the miracles seromned by : i-
menides, Aristeas, Hermotimus, Abaris, and Pherecydes. For oe
cydes, Theopompus is his source.6* The legendary material which
follows, about Pythagoras, is certainly derived from Aristotle: but
naturally the transitional sentence between Pherecydes and Pytha ane
which also separates the two sources Theopompus and ΝΞ Ὰ ᾿
supplied by the compiler, either Apollonius or his source Bolas It
corresponds to the Hellenistic conception of Pythagoras as βεβει ες
τῶν μαθημάτων, but does not give us any information about Aristotle’s
view of the matter; in his extant works he never connects Pythagoras
with μαθήματα. It would be superfluous to set this out in such detail
if it were not that the supposed fragment of Aristotle has been eited.
cven In very recent studies, as a basis for the assessment of Pythagoras’
contribution.*
The decisive passage is the introduction to Aristotle’s chapter on
the Pythagoreans in the Metaphysics: “ Contemporaneously with these
philosophers (Leucippus and Democritus) and before them, the so-called
Pythagorcans were the first to take up mathematics; ΤΡ advanced
this study, and having been brought up in it they thought its principles
were the principles of all things.’68 Taken in isolation, this passage
“δ Arist. fr. 191 = Ap. H.m. 6 = DK 14.7.
PGrHist 115F70; cf. ch. Π 3.
67 LT. - : :
Ἧς neha a pe EGP 97; eee Il 94 (who ingeniously combines this
; ‘ ; first lonian ἱστορίη, then Italian xaxoreyvin): Sarton, Hi ;
eee 25; Raven, PyEl 2; Mondolfo in ΖΜ 3$4; van der wae Astr. 5: es
Ὁ 1956, I 38; De Vogel, GPT ro, no. 22; Stapleton, Osiris 1958 44. The eotecet int
pretation: Heidel, AJP 1940, 8; Philip 23. oa
48 ; ἢ
ἈΠῈ τ Ἕ 98 5b23fF = DK 5884. The reading πρῶτον, which Christ adopted
oe recentiores, has against it the authority of the tradition (πρῶτοι is also found in th
second version of Alexander of Aphrodisias, on P. 37.12 Hayduck and Ascl. Met. 35 a
412
1, Pytnagoreans and the Foundations of Green Matnematics
could mean that the foundations of mathematics were laid by the
Pythagoreans, if they were the first to “take up and advance” it.
Still there is a chronological problem: What about Thales? Aristotle
is speaking of ‘*Pythagorcans,” and he dates them in relation to the
atomists.®® Thales, however, who is, in Aristotle’s view, the founder of
philosophy (Met. 983b20), was regarded as the patron saint of mathe-
matics even in the fifth century,’° and Eudemus ascribes to him some
special discoveries.
In order to understand why Aristotle passed over Thales in the passage
cited, one must consider the function of the passage in its context. The
objective of the first book of the Metaphysics is to demonstrate that
there are four first principles. The discoveries of earlier thinkers, in
spite of their many imperfections in detail, are all seen as moving, in the
developing history of thought, in the direction of this conclusion.
Whoever was “first” to bring forth a new view is carefully recorded.”
The “first” of the Pythagoreans is to be seen from this angle: they
“frst took up”? mathematics; this means that they were first to see the
relevance of the “principles of mathematics” to the general question
of “principles.” The question is not who invented mathematics, but
who connected mathematics with philosophy. From this point of
view, there was no occasion to name Thales; his mathematics had no
connection with his doctrine about water.
Inextricably entwined with this, in the passage of Aristotle, is
the psychological question of how the Pythagoreans came to their
system, which seemed as odd to him as it did to others.”* His answer
is that they devoted themselves so intensively to mathematics and
became so closely identified with it that they saw nothing in the whole
ee π- τ ς--Ξὸ- τς ee ee ee
also Bekker, Diels, Ross, Jaeger).—mpa@rot goes with ἁψάμενοι; cf. ἐξ ἀρχῆς ἁψάμενοι,
984228; Gen. corr. 320b34.—The sentence is cited, for example, by Zeller I 405 n. 2;
Heath, Math. 166; van der Waerden, Hermes 1943, 163, and often. We may add a passage
from a lost writing of Aristotle (lam. Comm. math. sc. p. 78.8ff; cf. above, ch. I 2, n.
112, and on the text, below, ch. VI 3, n. 1): οἱ δὲ Πυθαγόρειοι διατρίψαντες ev τοῖς
μαθήμασι καὶ τό τε ἀκριβὲς τῶν λόγων ἀγαπήσαντες, ὅτι μόνα εἶχεν ἀποδείξεις ὧν
μετεχειρίζοντο ἄνθρωποι, καὶ ὁμολογούμενα ὁρῶντες τὰ περὶ τὴν ἁρμονίαν ὅτι δι᾿ ἀριθμῶν,
ὅλως αἴτια τῶν ὄντων ταῦτα φήθησαν εἶναι καὶ τὰς τούτων ἀρχὰς...
69 Above, ch. I 2, nn. 099-00.
70 Below, n. 83. \
τι Cf. the constantly repeated πρῶτος and similar expressions, Met. 983b6, b20, b29,
98427, br8ff, b23, 985a8, a2of, 986b21.
72 Cf. Met. 986b24, on Xenophanes, and especially 987a29ff, on Plato (with the same
ingressive aorist φήθη 987b19, as 985b26). On ἐντραφέντες, cf. Pl. Tim. 19d. The balance
of the sentence, projected in the use of re-«at is disturbed by the intrusion of the psycho-
logical point of view and the interpolation of ἐντραφέντες . . . instead of, for example,
something like ταῦτά τε προήγαγον καὶ τὰς τούτων ἀρχὰς τῶν ὄντων ἀρχὰς ἔθεσαν.
413
VIE PYTHAGOREAN NUMBER THEORY
world but numbers.” "Thuy for Aristotle the Pythagoreans’ number
theory is a by-product of their mathematical studics, This view has
found adherents,” but is by no means self-evident. There is plenty of
number speculation which does not presuppose mathematics: and we
know from Aristotle’s own words that nonmathematical associations
were influential in forming the Pythagorean numerical cosmology.”
That it was mathematics of the deductive type which preceded and
produced the number philosophy is a psychological conjecture of
Aristotle, which the historian is not obliged to accept. We may credit
Aristotle's statement that there were certain Pythagoreans who achieved
succcss in mathematics’ and that there did exist a Pythagorean philo-
sophy of number. But as far as our information from Aristotle goes, it
remains an open question when the mathematical achievements
of these Pythagoreans occurred and how they are to be placed in the
general development of Greek mathematics. His concern was the
history, not of mathematics, but of philosophy.
A statement by Aristotle’s pupil Aristoxenus is obviously influenced
by his master: “Pythagoras seems to have honored, most of all, the
study of numbers, and to have advanced it in withdrawing it from the
usc of merchants and tradesmen, likening all things to numbers . . .77
The “advancement” of mathematics, and number philosophy, is here
moved back from the Pythagoreans to Pythagoras, and also scen in the
light of the Platonic demand that λογιστική should not be carried on
8 The same basic idea is expressed in a somewhat more friendly spirit in the parallel
passage cited above, n. 68: the Pythagoreans like the exactness of mathematics, see its
application in music, and by this route arrive at their number theory.
* Zeller, Vortr. u. Abh. 39: “From these mathematical and scientific studies they
proceeded to develop a set of beliefs about the nature of things in general”; Sarton,
Hist. 204: “In order to develop a mystical theory of numbers it was necessary first of all
to obtain a sufficient knowledge of them”—as though elementary calculation did not
provide plenty of material for numerical speculation, especially considering the existence
of Babylonian techniques of computation.
> Met. 985b29: ὅτι τὸ μὲν τοιονδὶ τῶν ἀριθμῶν πάθος δικαιοσύνη, τὸ δὲ τοιονδὶ ψυχὴ
καὶ νοῦς, ἕτερον δὲ καιρός... See below, ch. VI 4.
© This is confirmed by the details Eudemus gives; and Hippasus may also be recalled
in this connection (below, ch. VI 3).
” Aristox. fr. 23 = DK 58Bz2, according to Stob. 1 prooem. 6 ἐκ τῶν ᾿Αριστοξένου
περὶ ἀριθμητικῆς. Wehrli (54) doubts the title—Stobaeus’ whole introduction 1S περὶ
ἀριθμοῦ---αηά opines that “in the existing formulation, of an elementary introduction to
numerical concepts, fr. 23 is not from Aristoxenus.” One has the impression of an
abbreviated excerpt; but the introductory sentence, with its meticulous formulation,
looks like an exact quotation. The name of Pythagoras is cited with an air of introducing
a new clement. Wehrli’s assignment of the fragment to the book Περὶ Πυθαγόρου καὶ
τῶν γνωρίμων αὐτοῦ is conjectural. Both Frank (260 n. 1) and Wehrli (54) point out the
relation to Aristotle, as well as the connection between Aristoxenus’ further statement,
ot δὲ ἐκ τῶν θείων περιφορῶν ἐπινοηθῆναι (sc. φασὶν τὸν ἀριθμόν) with Ερίη. ογᾶς.
414
ι. Pythagoreans and the Foundations of Greek Mathematics
ἰδιωτικῶς .. οὐκ duty οὐδὲ πράσεως χάριν ὡς ἐμπόρους ἢ ial
μελετῶντας 15 It is worth noting that the subject is “numbers,” that is,
arithmetic, and not geometry. The fact that Aristoxenus uses the word
“honor” (τιμᾶν) shows that he was familiar with the nonmathematical
aspects of the Pythagoreans’ use of numbers. We sec the elements
from which the picture of the Great Master is taking shape. The way
in which the name of Pythagoras could acquire a truly mythical force
in such a context is illustrated by another report of Aristoxenus: “He
was also the first to introduce weights and measures among the
Greeks.”’’® In an egregious anachronism—as if the Homeric heroes
were still unable to measure or weigh anything!—Pythagoras is
classed with Prometheus and Palamedes.*°
The apparently ancient reports of the importance of Pythagoras and
his pupils in laying the foundations of mathematics crumble on touch,
and what we can get hold of is not authentic testimony by the efforts
of latecomers to paper over a crack, which they obviously found
surprising, by the use of various kinds of reconstruction and rein-
terpretation. On the other hand, there are ancient and unassailable
indications of a Greek mathematics antedating Pythagoras and quite
outside his sphere of influence. |
To be sure, the tradition about Thales has been the subject of
vigorous controversy ;*1 but in any case it was already firmly established
in the fifth century B.c.8? Before the entrance of the Thinking-shop
in Aristophanes’ Clouds stand statues of Astronomy and Gromeny,
The mention of geomctry leads to that of the world map (γῆς περίοδος),
which points unmistakably to the tradition of Ionic ἱστορία going back
at least to Anaximander (Nub. 201-216). Socrates 1s described by his
pupil as scattering ashes over a table and bending a cooking spit into
78 ῬῚ Rep. 5250.
79 Fr. 24 = D.L. 8.14. . .
80 Palamedes: Aesch. fr. 303 M., Soph. fr. 432 P., Gorg. Pal. 30; Pheidon of —
RE XIX 1943; Prometheus: Aesch. PV 459, and also Pl. Phib. 16c, where the inventor o
“re ' i Prometheus.”
the philosophy of number (above, ch. I 4) is called a
δἰ See τ τὴ HScH 54ff, Géom. 89ff; Heath, Math. I 128ff; van der Waerden, SA
85ff; esp. von Fritz, ΑΒΕ 1955, 77ff, and Becker, MD 37ff. Guthrie (I 52ff, 217ff), von
- Fritz, Becker, and Wehrli (Eudemos 115) have returned to a more positive evaluation of
+
the tradition about Thales, while the extreme skepticism of Burnet, EGP asf, and eae
(Essai 175ff) has been taken up again by Ὁ. R. Dicks, Thales,” CQ phate gee 2
IDicks, however, ignores the arguments of von Fritz and Becker, as well as . e pa =
in Aristophanes. On Thales’ alleged prediction of an eclipse of the sun, see Neugeba
HixSe 142. | ᾿ ;
2 See 13, Snell, “Die Nachrichten iiber die Lehren des Thales und die Anfange der
griechischen Philosophic- und Literaturgeschichte,” Philologus 96 obama he
determined that the intermediary source was Hippias (D.L. 1.24 - DK 8687).
415
ΡΥΤΗΛΟΟΒΠΑΝ NUMBER THEORY
shape to serve as a pair of compasses (διαβήτης), meaning of course
that he was going to make a geometrical diagram—and under cover of
this activity he steals a coat, ὁ Why do we go on admiring old Thales?”
cries Strepsiades (τί δῆτ᾽ ὀκεῖνον τὸν Θαλῆ θαυμάζομεν;). This shows
that for the Athenian public ‘Mhales and geometry belong together;
and this is also evident from another passage in Aristophanes. Meton
arrives in Cloudcuckooland; “I want to survey the air for you—
γεωμετρῆσαι βούλομαι τὸν ἀέρα suiv.”’ He will apply his ruler, “50
that the circle may become a square,” which will look like a star. .
All Pisthetacrus can say is “The man’s a Thales” (ἄνθρωπος Θαλῆς) 88
Eudemus gives detailed reports about mathematical propositions,
proofs, and constructions that Thales was supposed to have discovered, 88
he distinguishes between tradition and deductions of his own,
even recording an archaic locution used by Thales.®* This implies that
there was a book, available to him or his authority, ascribed to Thales.
The book in question must be that On the Solstice and the Equinox.87
Whatever the situation may be with regard to authenticity, there
obviously existed, in the sixth century, Ionic technical writings on
problems of astronomy and the calendar;88 already with them,
geometrical concepts—circles and angles—seem to take the place of
5 Ar. Av. 995ff, 1009. These references to Thales are not included in DK. Cf. B.
Gladigow, Hermes 96 (1968) 264-275; he connects the compasses with Thales’ measure-
ments of distance (A20).
δὲ DK 11A20; Eudemus fr. 134 = Procl. In Eucl. 352.14 (the congruence of triangles
which have one side and two corresponding angles equal); fr. 135 = Procl. In Encl.
299.1 (equality of the vertically opposite angle); probably from the same source, Procl.
In Eucl. 157.10 (bisection of the circle by the diameter) and 250.20 (the equality of the
base angles in an isosceles triangle).—Hieronymus of Rhodes fr. 40 W. = D. L. 1.27
(with Plin. HN 36.82, Plut. Conv. sept. sap. 147a, DK 11A21): measurement of the height
of a pyramid.—The attribution to Thales of the theorem that the angle in a semicircle
is a right angle is problematic (Pamphila ap. D.L. 1.24f; Heath, Math. I 134ff). It is
closely connected with the theorem on the sum of the angles of a triangle, for which,
according to Eudemus fr. 136, the Pythagoreans were the first to find the proof.
"Fr. 134: τὴν yap τῶν ἐν θαλάττῃ πλοίων ἀπόστασιν δι᾽ of τρόπου φασὶν αὐτὸν δεικνύναι,
τούτῳ (τῷ θεωρήματι͵) προσχρῆσθαι... ἀναγκαῖον, From this it appears that what he
found in the tradition was a specific method for determining distances (φασὶν), and
Eudemus himself decides what are the underlying mathematical propositions.
ΜΔ Procl. In Eucl. 251.1 (equality of the base angles): ἀρχαικώτερον δὲ τὰς ἴσας (γωνίας )
ὁμοίας προσειρηκέναι, cf. Becker, QSt 4, 151f; DK I 486.36ff (Nachtrage); von Fritz,
ABCs 1959, 48f.
"7 Περὶ τροπῆς καὶ ἰσημερίας, D.L. 1.23, Suda s.v. Thales; taken by D.L., probably
through a misunderstanding, as two works. Eudenius (fr. 145; cf. 144) did in fact ascribe
the discovery of the inequality of the seasons to Thales. Gigon (Ursprung $5) thought of
the Ναυτικὴ ἀστρολογία, but the word γωνία will not fit into a hexameter poem (DK
111).
ὅδ Thales’ rivals are Phocus of Samos (D.L. 1.23) and Mandrolytus of Priene (Apul.
Flor. 18 --. DK 11Atg); and in the same tradition are Matricetas of Methymina and Cleo-
stratus of Tenedos (DK 6).
416
1. Pytnagoreans and the Foundations of Green Mathematics
Babylonian calculation. ‘The Babylonians do not use the concept of the
angle.#® Becker has shown that all of the propositions attributed to
Thales can be derived simply from considerations of symmetry,"" and
von Fritz points out that the method assumed, that of as ae
(ἐφαρμόζειν), was consciously avoided in later Greck geometry. :
With Thales the point is still a graphic or perceptible “showing
(δεικνύναι)."2 But in this perspicuity itself there is a new element by
contrast with the Babylonian “recipes.” It is in the perceptible figure
that mathematical propositions become clear in all their generality
and necessity: Greek geometry begins to take form. | |
Thales’ name was the only one to remain familiar ;* aside from him
we can discern the specifically Greek innovation in geometrical thought
only through its effects. The world of Anaximander is connie’
on severely mathematical ratios” ;® it is “essentially geometrical.
Cosmology is subject to mathematical logic; the earth must hang free
in the middle because it is equidistant, in every direction, from the
‘“wheels”’ of the stars. His statements about the size and distance of the
heavenly bodies, in which geometry extends its purview to the whole
universe, presuppose at least some knowledge about geometrical
proportion in the correlation of distance with true and apparent
magnitudes.%* Anaximander relies on the accuracy of geometry in
matters beyond the range of any kind of verification—in its application
89 Babylonian mathematics has a measure of inclination (Sagal, = mad en aaa
responding to the cotangent function) but not the concept of an angle (Becker, Grd.
27, MD το, 37).
90 MD 37ff. Ε :
Ἰὼ BG i assim. .
32 ΠΡ RES aT als mathematischer Terminus fiir ‘beweisen’,” Maia
ἘΠ hard ‘a make out anything about the brother of Stesichorus menage’ as a
geometer by Hippias (Procl. In Eucl. 65.12 = DK 86812 = Eudemus fr. 133, ἐν ae
of Hero Deff. 136.1 p. 108.12 Heiberg and Suda s.v. Stesichorus). The ee ‘
the name is probably Mapéprios (as Hero; the Suda has Mapeprivos, MS: a τὰν i
Apept', apogr. ᾿Αμέριστος corr. Μάμερκος); Mapépriov is the name ofa ἐν "" Ὄ pee
near Rhegium (Oldfather, RE XIV 952). Thus combinations with oe ost . ae τὰ
father (D.L. 8.1) or Mamerkos the son of Pythagoras (Plut. Aem. Paul. 1,1 sale ; :
p. 22 L.; M. Detienne RHR 152 [1957] 142) are built on sand, especially since Stesichorus
is generally agreed to have lived in the first half of the 6th century.
94 idei Eng. tr.). .
95 eats re ae ae 7 : “ Anaximander is ancestor ... also to a geometric
philosophy usually associated with the name of Pythagoras”’; cf. 77, 92ff, The assertion
in the Suda that ‘‘in general, (Anaximander) drew up the blueprint for geometry
(s.v. Anaximandros=DK 12A2:... ἰσημερίαν εὗρε καὶ τροπᾶς καὶ ὡρολογεῖα Re ;
γνώμονά τε εἰσήγαγε καὶ ὅλως γεωμετρίας ὑποτύπωσιν ἐδειξεν) is a cruder version ©
Favorinus ap. D. L. 2.1f: εὗρεν δὲ καὶ γνώμονα ... τροπάς τε καὶ ἰσημερίας σημαίνοντα
καὶ ὡροσκοπεῖα κατεσκεύασε καὶ γῆς καὶ θαλάσσης περίμετρον πρῶτος ἔγραψεν.
96 Above ch. IV I.
417
fo cosmic proportions amd alse ΠῚ contradiction τὸ appearance,
which suggests that the sun is about as large in diameter as the width
ofa human foot.
The concept of geometrical similarity is also the precondition for
Anaximander’s attempt to construct a map of the world. And geo-
metrical thought is still in the saddle in the work of Hecataeus, mocked
by Herodotus for representing the carth as circular (quartered by the
Nile and the Danube) and depicting less well known countries like
Libya as built of rectangles and squares. 9”
In the time of Polycrates, the architect Eupalinus of Megara showed
the same kind of bold and confident geometrical thinking. The water
supply for the city of Samos was routed through a mountain, in a
conduit about a kilometer long, a technological accomplishment at-
tested by Herodotus (3.60). The historian does not mention, however,
the fact which appeared when the tunnel was rediscovered in 1882,
that the tunnel had been dug from the two ends simultaneously, 98
the two galleries meeting, with slight inaccuracy, in the middle. Even
though this undertaking did not require complicated geometry,
nothing more in fact than relatively simple measurements, the daring
character of the enterprise remains. The geometer relied on his
στοιχεῖα even where the facts could not be directly checked, and the
accuracy of his plan could only be tested after years of work. When
Hezekiah had a similar conduit constructed for Jerusalem, about
700 B.C., the direction was checked and corrected by means of a series
of vertical test shafts, so that the channel when completed ran a zigzag
course. 99
We know the names and accomplishments of a whole series of
contemporary technologists They point to a phenomenon only
scantily attested in the literary tradition, which contributed to the
development of mathematics, namely the striking advance in technolo-
logy, to be seen above all in temple architecture from the end of the
δ᾽ See Jacoby’s note on FGrHist 1F36 (northern Libya, Hat. 4.168ff; Thrace frr.
146-183; Scythia ἔτ. 184-190) Hdt. 4.36, 2.16, 33f. On his dependence on Anaximander,
Kahn 82ff.
88 E. Fabricius, MDAI 9 (1884) 165-192; Burchner, REI A 2189 s.v. Samos; W. Kasten-
bein, AA 1960, 178-198: J. Goodfield and 5. Toulmin, Isis 56 (1965) 46-55. Goodfield
and Toulmin argue persuasively that the direction of the tunnel was regulated by means
of a row of stakes running across the mountain, against van der Waerden, who cited
Hero’s Dioptra 15 (SA 102-105). The poles set in line would probably have been called
στοιχεῖα (ct. στοιχίζω, περιστοιχίζω, διαστοιχίζομαι).
99 Van der Waerden, SA 10 2.
109 See Sarton, Hist. 188ff: Theodorus and Rhoecus of Samos (inventors of the hollow
casting of bronze), Chersiphron of Cnossus and his son Metagenes (movement of columns:
see Vitr. De arch. 10.2.11f), Mandrocles of Samos (first bridge of ships, 514 B.C.).
418
t, Pythagoreans and the Voundations of Greek Mathematies
seventh century, Certun technical terms of geometry clearly are
borrowed from architecture, like yeoréa, bp0h γωνία, and τε τρείγενον, "9
as well as the geometrical tools κανών, γνώμων, and διιβήτην "Ἂς
early as Theognis and Simonides we find these concepts or words used
as symbolic of impeccable truthfulness and accuracy.'* Is this elevation
of ideas, from the realm of craftsmanship to the plane of the symbolic,
Pythagorean? Pythagoras of Samos came from a milieu in) which
technological and geometrical thinking was in its heyday. It is un-
thinkable that he took no notice of it; but there is no way of knowing
whether he made any contribution to it himself. The carcer of Greek
geometry began before Pythagoras’ time, and there is no warrant for
supposing that cvery trace of mathematical and geometrical thought
in early Greece is eo ipso Pythagorean.1%
161 γωνία used of buildings, Hdt. 1.51 et saep.; IG XII 7.1; γωνιαῖοι λίθοι and the like
in building accounts, IG I? 372.19, 161, 313.86, 373.80. See A. Debrunner, IF 60 (1949)
38-46. It is not entirely clear whether the vowel in γωνία shows compensatory lengthening
or the lengthened grade (as Schwyzer thinks, I 358). If it is compensatory lengthening,
Debrunner would like to refer to the Pythagoreans of southern Italy, though he must
admit that for this area the compensatory lengthening to w is not firmly attested. We
can regard it as certain that architecture before Pythagoras did not get along without the
term τετράγωνος (πλαίσιον is only the brick mold); and perhaps one ought not to dismiss
as readily as Debrunner Hesychius’ gloss γῶνορ, γωνία. Ad«wves.—According to Proclus
In Remp. Il 26.18 and Hero Deff. 15, the Pythagoreans called the angle not γωνία but
yAwyis (elsewhere used of the feathering of an arrow: τρίγλωχιν θιστόν, Tl. ς.303,11. 507,
Soph. Trach. 681). There are examples of τρίγλωχις, “‘3-cornered,” Pi. fr. 322, Callim.
fr. 1.36; of τετράγλωχις, “4-cornered,” Leonidas A. P. 6. 334.3; and an expression
ἰσογλώχινι τριγώνῳ, Nonnus Dion. 6.23. This evidence would seem to show that yAwyis
is a poetical term, used as a substitute for the unmetrical ywvia. If ancient Pythagoreans
had in fact used it, the whole development of Greek geometry was unaffected: Procl.
In Encl. 130.8 etc. = Philolaus A14 uses γωνία as casually as Eudemus fr. 141 = Archytas
Al4. ᾿
105 The restriction of this word’s sense—from “quadrangle” to “rectangle,” and usually
“square’’—is a development resulting from the usage of masons. For τετράγωνον in
the context of architecture, see Hdt. 1.178, 181, 2.124; IG I? 313.401. ;
108 According to Plin. HN 7.198, the inventions of Theodorus of Samos included
“norma, libella, tornus, clavis.”” Technologists in particular had connections in the
Orient. Not only did they build bridges of ships for Darius and Xerxes, but there were
Greek masons at the court of Cyrus I (G. M. A. Richter, AJA 50 [1946] 15-30). ;
104 Theog. 805f: répvov καὶ στάθμης Kat γνώμονος ἄνδρα θεωρὸν εὐθύτερον χρὴ (έ ) μεν,
Κύρνε, φυλασσόμενον (cf. 543); Simonides 542.3 Page: ἄνδρ ἀγαθὸν μὲν ἀλαθέως γενέσθαι
χαλεπόν, χερσίν τε καὶ ποσὶν καὶ νόῳ τετράγωνον, ἄνευ ψόγου πεΤο μεύον" (like a
building stone with which the supervisor or foreman can find no fault. Cf. the “corner-
stone” of Pss. 117.22, Isa. 28.16, Matt. 21.42, Acts 4.11). A dedicatory inscription from
the neighborhood of Sybaris (Bull. Epigr. 1967, no. 697, 6th century B.C.) has a noteworthy
expression about a votive offering Εἰσοίμ) μᾶκός τε πάχος τε (ἃ cube?). A pot by the
Cage Painter (Louvre G 318; E. Potticr, Vases antiques du Louvre I [1022] pl. 135;
ARV? 348.3; dated 480/470 B.c.) has a scene representing instruction in geometry:
a boy with a pair of compasses, and a γνώμων hanging on the wall. .
105 Frankel drew attention to the thought pattern of the “geometric mean” in Hera-
clitus—man is to god as ape is to man, etc. —and thinks there may be Pythagorean i=
fluence here (AJP $9 [1938] 309-337, in German IVF 253-283, esp. p. 205 n. $i similarly
Minar, CP 34 [1939] 337-340). The idea of proportion, however, antedates | ythagoras.
419
PL PYTHAGOREAN NUMBER THEORY
Callimachus makes play with the crossing of the traditions about
Thales and Pythagoras in the well-known passage from the Jambi, in
which an Arcadian is to give a bowl “τὸ the wisest.” First he comes to
Thales, who is “scraping the carth and drawing the figure which the
Phrygian Euphorbus discovered, the first of men to draw triangles and
oblique figures and the curved spiral, who taught us to abstain from
animal food... .”!8 Thales is concerned with geometrical propositions
discovered by ‘Phrygian Euphorbus,” which naturally means Pythag-
oras.!° This has been misunderstood to mean that some kind of
tradition actually ascribed such discoveries to Eurphorbus,!°8 or that
Pythagoras, as a youthful prodigy, founded Greek mathematics during
Thales’ lifetime.°® This makes the chronological inconsistency, to
which Callimachus so wittily alludes, disappear. He consciously
ascribes to Thales a knowledge that he could not have had, chronolog-
ically, but then, with the help of the doctrine of metempsychosis,
makes the impossible possible. Pythagoras introduced geometry from
Egypt to Greece and brought it to perfection; yet before Pythagoras’
time Thales was already famous as a geometer. Therefore Pythagoras
must have made his discoveries in an earlier incarnation.
From about the middle of the fifth century, it is clear that mathe-
matics is a center of intellectual interest. Almost all the important
thinkers are concerned with mathematical questions. Anaxagoras,
Hippias, Antiphon, and Hippocrates of Chios worked on the squaring
of the circle; this was a problem so widely known that it could be
106 Callim. fr. 191.$8-62 Pfeiffer. The words τὸ σχῆμα (“the figure”) may hold an
allusion to the Pythagorean theorem (Pfeiffer and Howald in the Artemis edition, 1955,
P. 320f). τρίγωνα καὶ oxadnvd (“triangles and oblique figures”) reminds of Eudemus fir.
136-137 (below, ch. VI 3). In the first part of verse 61 (καὶ κύκλον EA[ixa]) Diodorus
(10.6.4) has the unmetrical κύκλον ἑπταμήκη, POxy. VII rorr (p. 32; cf. p. 71), Hunt
reads ἐπὶ with A written above 7; according to Pfeiffer ἐπ is “‘satis certa.” Pfeiffer con-
siders ἔπαγε or ἔταμε, Maas suggests ἔπλασε. Diels’s conjecture ἕλικα (DK) has an import-
ant point in its favor: ἕλιξ is a technical term for the apparently “spiral”? movement of
the planets which results from the overlapping of the revolution of the earth and the
planets’ own movement (PI. Tin. 39a, Hermesianax fr. 2.86f Diehl, Ptol. A. P. 9.577);
and the description of the zodiac was ascribed to Pythagoras, in rivalry with Oenopides.
As the word νηστεύειν shows, Callimachus is not speaking of mathematics alone, but is
giving, in a concise form, a general characterization of Pythagoras’ teaching—and
therefore could certainly not leave out astronomy (cf. lines 54f, on the astronomy of
Thales). On the text of line 62, see H. Lloyd-Jones, CR 17 (1967) 125-127.
107 Above, ch. II 3.
™ Delatte, Vie 157f, makes Euphorbus, “ Good-Shepherd,” a Phrygian culture hero:
Rostagni agrees, Verbo 120 ἢ, 1, 240 ἢ. 1.
Lévy, Sources 4rf,
420
1, Pythagoreans and the Foundations of Greek Mathematics
used in comedy. Hippocrates of Chios was pioneer on the line of
thought which Archytas followed up in solving the problem of
doubling the cube.!"" Protagoras tried to refute the geometers on
principle by maintaining that their postulates never fit reality: there
is no visible line touching a circle at only one point." His disciple
Theodorus dealt with irrationality.“3 Democritus did a good deal of
work on mathematical problems."4 The statues of Geometry and
Astronomy at the entrance of Socrates’ ““Thinking-shop” are a clear
indication that before the end of the fifth century the branches of
mathematics, as such, had a firm place in the curriculum of the Sophists’
program of higher education.“ When Xenophon represents Socrates
as criticizing excessive enthusiasm about geometry, astronomy,
λογισμοί, and medicine, when they are carried beyond the needs of
practicality, this is not simply an anachronistic criticism of Plato.“
Hippias, as Plato tells us more than once, taught λογισμοὺς καὶ ἀστρονομίαν
καὶ γεωμετρίαν καὶ μουσικήν.7 And all these special fields are also
attested for Democritus. It follows that the quadrivium is not merely
a Pythagorean import brought by Plato from Italy. If, as in the later
tradition, it is to be regarded as Pythagorean," its influence must have
110 See Heath I 233ff; Becker, MD 93ff; Anaxagoras A38; Hippias fr. 21 (on which,
Becker, MD gsff, van der Waerden, SA 146, 191; Heath, Math. I 182, 226ff); Antiphon
fr. 13; Hippocrates (above, n. 10); Ar. Av. 1005.
11 DK 42.4; cf. 47Ar4.
112 DK 80B7 = Arist. Met. 997b32. Probably Democritus treated the same problem
in his book (Bril) Περὶ διαφορῆς γνώμης ἢ περὶ ψαύσιος κύκλου καὶ adaipns.
118 DK 43.4 = Pl. Tht. 147d; cf. above, ἢ. 12; below, ch. VI 3.
114 Democritus’ importance in the history of mathematics was expounded by Frank
(esp. 82), not without a certain amount of exaggeration. Archimedes gives him credit
for discovering how to determine the volume of a cone (DK II 174 n.). The fact that he
is not included in Proclus’ catalogue of geometers may be due to the prejudice of the
Platonists (as van der Waerden thinks, SA 91). Cf. the preceding note, and on the quad-
rivium, below, n. 118; on irrationality, below ch. VI 3.
116 Ar. Nub. 200ff. Strepsiades must also learn μουσική; he can already calculate interest.
116 Xen. Mem. 4.7.2ff.
117 Pl. Prot. 318d-e, Hp. ma. 28s5b, Hp. mi. 366c, 368e.
118 Geometry: Br1l, m,n, p; arithmetic: the title ἀριθμοί, Blo; astronomy: Briq—
B14; music: B1sc-B26a. Frank (10 and n. 23) sees in Democritus the founder of the
quadrivium; but Hippias was about the same age, and the systematic association of the
four subjects cannot be shown to be present in Democritus. "
119 Nicom. Ar. 1.3, with citations of ‘‘ Androcydes” and Archytas (fr. 1); a citation of
Pythagoras Th. ar. 21.7 (from the book περὲ θεῶν); “Cleinias,” ibid.; ps.-Archytas
p- 6.11 Thesleff. The quadrivium is also Pythagorean for Proclus, In Eucl. 35.21ff; but
Theo (16ff) speaks only of Plato. On Theodorus, above, n. 12. Morrison, CQ eae
203ff, recognizes that “higher mathematics, astronomy, geometry, and the theory of
numbers were all the subject of lectures at Athens at the close of the fifth century
(213), but assumes, for this reason, that Hippias and Theodorus “brought Pythagorean
mathematics to Athens” (203).
421
vie PYLHAGOREAN NUMBER THEORY
extended τὸ Hippias, Democritus, and. the Aristophanic Socrates,
It was the cultural influence of the Academy that brought the system
of the “four fields” to their position of special prominence;}?° the
only unequivocally Pythagorean clement is the arithmetization of
music theory and, to a degree, the elevation of number theory (“‘arith-
metic’) to an independent branch alongside geometry.121 All the rest
had been generally known and could assume their place in the scheme
of higher education without the help of any esoteric or Pythagorean
influence. This is the way Isocrates is looking at the matter when he
distinguishes between the traditional pair, γραμματική and μουσική
and the subjects that had emerged in his own time, namely sees i
ἀστρολογία, and διάλογοι ἐριστικοί.135 Here he js alluding to the sdk
tional system of the Academy.
Along with the appearance of the quadrivium comes the semantic
development, in which the word μαθήματα, “fields of study,” is
reduced to the branches of “mathematical” study, while παν
is restricted to mathematics proper (including astronomy). This usage
is not consistently followed by Plato and Isocrates until their later
works,'5 to be taken up then in the Epinomis and in the works of
Aristotle. In general Plato still uses μάθημα in its original, broader
scnsc. Thus it seems that the word “mathematics” became fixed only
in the time of the Old Academy. The question of the extent to
which Pythagoreans anticipated Plato, in the treatment of the four
branches and the development of the concept of the μαθήματα,
depends on the problem of the genuineness of the long fragment of
Archytas.14
In any case Greek geometry assumed its final form in the context
of the Old Academy. After Plato had placed an especially high valua-
tion on mathematics, and had fixed its position as a discipline of pure
thought (so that Protagoras’ objections became irrelevant), there came
29 Plato treats the four μαθήματα, already adding solid geometry, in Rep. 524dff:
“ee lso Leg. 747A, Epin. 990cff, 991c. On the development from Plato on, see Mera:
PINcopl 78ff; he does not go into the pre-Platonic period. ᾿
"1 Below, ch. VI 2.
12 sae fas 26; cf. Antid. 261, 264f., 266f, Bus. 23; on the relationship to Plato
acger, Paidcia 147 (217 Ger. ed.); K. Ries, Isokrates und Platon i | 2
Philosophia (Diss. Munich, 1050). onan ee tee
ay PI. Leg. 817¢, Isoc. Auntid. 261, 10, Panath. 27.
Above, ch. V1, ἢ. 46; on the mathematici, above, ch. Il 5. For a Pythagorean origin
of the μαθήματα concept, B. Snell, Die Ausdriicke fiir den Begriff des Wissens in der
vorplatonischen Philosophie (Berlin, 192 ff; Heath. Math ἡ
1960, 2of,. 4) 77ff; Heath, Math. I 11; von Fritz, SBMii
422
ι. Pythagoreans and the Foundations of Greek Mathematics
an unprecedented development of these studies. Aristotle writes,
“Those who concern themselves with geometry and calculation and the
other sciences have from small beginnings made by now such progress
in a very short time as no other ficld has made in any of the arts,"
But even in the age of Anaxagoras, Protagoras, and Hippocrates of
Chios, we notice that mathematicians are no longer concerned merely
with individual problems, but are working on fundamental ones.
The problems of squaring the circle and doubling the cube are not
soluble by ordinary geometrical means, with the use of ruler and
compass; and the fact of irrationality can only be deduced, or “shown”
in a logical argument, never made immediately perceptible. Here
geometry freed itself from its bondage to the needs of practicality.
For the needs of everyday life, there were already plenty of approxima-
tive values to use in the calculations involving roots and the dimensions
of the circle. In the geometry of the fifth century, however, it was
obviously recognized that these were merely approximations, and that
there is a basic difference between these and any exact solution resting
on proof; it was understood that the task of expressing a magnitude
like ./2 in whole numbers, or expressing the length of the diagonal in
terms of that of the sides of a square, was not only “not yet”’ solved,
but in principle insoluble. Mathematical logic and deductive proof go
beyond what is perceptible, and this is what carried Greek geometry
far beyond its predecessors, no matter how suggestive, in the oriental
cultures.
If a basic influence was exerted by Pythagoras or by Pythagorcans
on Greek mathematics, this would have had to take place in the period
between Anaximander, Cleostratus, and Hecataeus, on the one hand,
and Anaxagoras, Oenopides, and Hippocrates of Chios on the other.
What ensued was, in the first place, the development of detail and, in
the second, a metaphysical and logical undergirding in which the
125 Acad. Ind. Herc. p. 15ff Mekler (cf. below, ch. VI 3, ἡ. 86); Procl. In Eucl. 66.8 =
Eudemus fr. 133, where no fewer than 11 mathematicians between Plato and Euclid are
named—and we may add Polemarchus and Helicon of Cyzicus, Amphinomus, Bion of
‘“Abdera, Aristotherus, Autolycus, and a certain Aristaeus. Even if Plato was not a pro-
fessional mathematician, his philosophy seems to have provided the decisive breakthrough
in the establishment of mathematics, and at the same time to have directed the attention
of philosophers, even more than had been the case, toward mathematics. See above n. 6.
On the motto dyewpérpnros μηδεὶς εἰσίτω, see H. D. Saffrey, REG 81 (1968) 67-87.
126 Fr. ς == Jam. Comm. math. se. p. 83.13ff (with Aristotelian origin guaranteed by
Cic. Titsc. 3.69): τοσοῦτον δὲ viv προεληλύθασιν ἐκ μικρῶν ἀφορμῶν ev ἐλαχίστῳ
χρόνῳ ζητοῦντες οἵ τε περὶ τὴν γεωμετρίαν καὶ τοὺς λόγους καὶ τὰς ἄλλας παιδείας, ὅσον
οὐδὲν ἕτερον γένος ἐν οὐδεμιᾷ τῶν τεχνῶν,
Vai FP ATAUUAGAIN INUMBOR PfFIbUAT
Pythagorcans, according to the testimony of Aristotle, had no part:
τὰ γοῦν θεωρήματα πρυσίπτουσι τοῖς σώμασιν ὡς ἐξ ἐκείνων ὄντων τῶν
ἀριθμῶν."37 In that carly period there docs seem to be a certain gap,
coinciding with the turbulent period of the Persian Wars. There were,
nevertheless, direct connections between the older Ionic school and
Anaxagoras, and, as Anaximandcr’s pupil Cleostratus was an astrono-
mer, so was Oenopides of Chios, the pupil of Anaxagoras and compat-
triot of Hippocrates. According to Eudemus, Oenopides worked out,
for the first time, certain elementary geometrical constructions “since
he believed that this would be useful for astronomy.’’?8 The signifi-
cance of these apparently simple accomplishments seems to lie in the
fact that in them geometry consciously restricts itself to the tools of
compass and ruler.2° Oenopides is still not doing mathematics for its
own sake; but in the demand for precise construction, and the concept
of the πρόβλημα,130 he represents a definite forward step in the direction
of “pure” theory. Hippocrates, too, was concerned with astronomy,}31
but he also wrote a book called Στοιχεῖα. If there were more reliable
evidence about the discovery of pure theory by Pythagoras than the
philosophia anecdote of Heraclides and the supposed Eudemus,122
then one would have to recognize the true accomplishment of the
Pythagorean school through its influence here in Oenopides and
Hippocrates, and one might speak with perfect justification of the
Pythagorean basis of Greek mathematics. But there is another
solution.
The attempt at purely logical argumentation, a systematic progres-
sion from one thought to another, and the advancement of proofs and
conclusions in conscious contradiction to the evidence of the senses
make their first appearance in Parmenides. Kurt von Fritz has shown
how νοεῖν, which previously meant an intuitive comprehension, first
became logical “‘thinking’’ in Parmenides.18 “They were led to
127 Arist. Met. 1083b18.
138 Qenopides, DK 41.13 = Procl. In Eucl. 283.4 (constructing a perpendicular to a
straight line through a given point); DK 41.14 = Procl. In Eucl. 333.1 = Eudemus fr.
138 (to construct an angle equal to a given angle at a given point on a straight line);
the first notice also obviously comes from Eudemus.—CCAG VIII 3, p. 95.12:
πρῶτος. ... τὰς ἀστρολογικὰς μεθόδους ἐξήνεγκεν eis γραφήν.
#29 ψοῃ Fritz, RE XVII 2266. Every surveyor has a simple instrument for measuring
right angles (basically the γνώμων). Oenopides in fact uses the expression κατὰ γνώμονα
for “vertical” (DK 41.13), but he constructs the angle without mechanical help.
180 DK 41.12.
131 Cf. above, ch. IV 1, n. 77. On στοιχεῖα, Burkert, Philologus 1959.
182 On Heraclides, Burkert, Hermes 1960, 159ff; on Eudemus fr. 133, above, ἢ. 62.
183 Von Fritz, CP 40 (1945) 223-242, 41 (1946) 12-34, esp. 1945, 241.
424
ἢ, FP YRAQOFEAN ANG [NO POUNGAHOMS OF VETEER ἐν ΘΕ ΠΕΊΘΗΙ
transcend sctise perception and to disregard it on the ground that
‘one ought to follow the argument’ ’—thisis Aristotle's characterization
of the Eleatics,™ and deoAovJeiv τῷ λόγῳ is the distinctive feature of
Greek mathematics as early as Hippocrates. ‘La mathématique grecque
restera plus ou moins éléatique jusqu’au bout” is Abel Rey’s way of
putting the matter (148), and more recently Arpdd Szabé, in a number
of publications, has shown the dependence of Greek mathematics on
the thought of the Eleatics.18° The connection of geometry, and
especially that of Hippocrates of Chios, with the logic of the Eleatics is
obvious. The Eleatic thought pattern of the differentiation of cases
dominates Hippocrates’ treatment of the problem of the quadrature
of lunes; the exterior arc of the lune is either greater or less than a
semicircle, or equal to it, so that all possible cases are exhausted.186
Zeno’s methods of proof, the reductio ad absurdum and the regressus
in infinitum, are basic to all the proofs about irrationality.%’ In the
logic of the Eleatics we find the factor that brought about the advance-
ment of mathematics, in the hiatus between Anaximander and Anaxa-
goras; there is a similar development in natural philosophy between
the early Ionians and the post-Parmenidean thinkers. Anaxagoras,
Empedocles, and Leucippus, each in his own fashion, achieved an
advance by the combination of Ionian beginnings with Eleatic ontology
and patterns of thought. There followed then the playful intellectual
diversions of the Sophists. That which had emerged from everyday
activity and bold speculation in the sixth century, was discussed,
sifted, and gradually brought into a logical system. In the process,
natural philosophy drifted into a dilemma between eclectisism and
skepticism, but geometry took on its typical Greek form.
A contrary interpretation would have Parmenides deriving his style
of argumentation from Pythagorean mathematics: “The method of
184 Gen. corr. 325413; cf. Cael. 298b22: the Eleatics were the first to believe in a
Being uncreated and immobile, though not (as in Plato) distinguished from the
perceptible.
135 4A 1955, 67ff; ‘‘Wie ist die Mathematik zu einer deduktiven Wissenschaft ge-
worden?” AA 1956, 109-152: “4είκνυμι als mathematischer Terminus fiir “beweisen,’”
Maia 10 (1958) 117ff; “Die Grundlagen in der friihgriechischen Mathematik,” StudIt 30
(1958) 1: “the historically necessary prerequisite for the earliest Greek mathematical
science”’ is “‘the Eleatic philosophy.” All the same, Szabé does not free himself from the
communis opinio on Pythagorean mathematics, but accepts Becker’s ‘‘doctrine of even
and odd,” and places the Pythagoreans, as “τῆς first representatives of deductive mathe-
matics” (Maia 1958, 130), between Parmenides and Hippocrates. Were the Pythagoreans
the only ones who could learn mathematical logic from Parmenides?
136 Above, ἢ. 10; cf. Reinhardt, Parm. 35ff, 64ff; above, ch. III 2, ἢ. 104.
137 Cf. below, ch. VI 2, n. 47; VI 3, n. 81.
425
EEE EE EL OES ἢ Oe ἦν ΨΚ ὮΡ eer Ὁ Υ'Ἱ δα δ δ Te αὶ ἢ. δ
reasoning he imported into philosophy is the method of geometry, ᾽138
Certainly; but does Parmenides “import” someone else’s results? The
price at which the Pythagorean origin of mathematics is saved, in this
interpretation, is tremendous, Are we to suppose that Parmenides merely
applied a previously developed method to a new concept, τὸ édv? Only
if Parmenides himself gave some indication of the fact, or if other
testimony gave a clear indication of the existence of Pythagorean
mathematics before Parmenides, would this thesis be acceptable. It
would greatly diminish the originality and the basic importance of
Parmenides in the growth of Greek philosophy, which has been
generally recognized since the work of Reinhardt. In Parmenides
himself there is not a word that points toward the field of mathe-
matics—what a contrast with Plato and Aristotle! Further, if it can
be shown that Zeno, in his manner of argument, holds close to the
ideas of his teacher,!8® and if Zeno also, in his manner of expression,
shows no dependence on mathematics, then there is no occasion to
interpolate a ““Pythagorean mathematics” between the two! In
discussing Being, Parmenides discovered the independence of thought;
and deductive mathematics as well as logic took rise from. this
beginning; from the point of view of the development of thought,
ontology is prior to the formal schematism.
Greek mathematics did not emerge from the revelation of a Wise
Man, and not in the secret precinct of a sect founded for the purpose,
but in close connection with the development of the rational Greek
view of the world. Taking its departure from “‘ measurement,”
gcometry becomes a component of natural philosophy, φυσιολογία.
In one of these realms as well as the other, Thales and Anaximander,
Parmenides and Zeno are the important names, and even Oenopides
and Hippocrates are φυσικοί in their astronomy. Then, earlier than
other fields, geometry and astronomy become the domain of spe-
188 Cornford, PrSap 117; cf. PlParm 29; Rey has the same judgment, 191; cf. Cherniss,
JHI 1951, 336: “The example of mathematical proof he [Parmenides] learned from them
[sc. the Pythagoreans] may have determined the rigorously deductive form of his argument
and may have been the origin of his confidence in this method.” See also T. B. L. Webster,
Greek Art and Literature 700-530 B.c. (London, 1959) g1ff. But there is no proof given
that deductive geometrical proof was characteristic of the Pythagoreans before Parmen-
ides. See above, n. 92.
199 Above, ch. ΠῚ 3, n. $5.
140 Though Szabé does so, once more, in AA 1955, 83ff. That Zeno speaks of adding,
taking away, and προέχειν does not imply special mathematics; and the role of bisection
is common enough. There is a problem of proportionality in the millet-seed argument
(A29), but this of of dubious authenticity, and in any case is not from Zeno’s book (cf.
above, ch. III 3, n. 47).
426
cialists, because thet: imereasing: complexity demanded a spree slic ally
mathematical talent, which is not dependent on one's origi or one’s
membership in any kind of school. At the same time, the Sophists
brought about a division, since the exactness of mathematical results
was more and more obviously in contrast with the uncertainty of
φυσιολογία. Thus in Plato’s time mathematics was already the model
science, which even skeptisicm had to take seriously. Individual
Pythagoreans had some part in this development, but in its “essence
mathematics is not Pythagorean but Greek.
2. PYTHAGOREAN ARITHMETIC
That Pythagoreans were much concerned with numbers is established
fact, but it is a question, in what sense this activity can be called
mathematics; for the paradigmatic form of Greek mathematics, as a
deductive system based on axioms, is geometry. On the other hand,
there is one remarkable type of arithmetic that appears exclusively in
the Pythagorean tradition, in which numbers are represented by figures
made with counters or pebbles, ψῆφοι.1 Aristotle knows of triangular
numbers”’;? and the “perfect” number το, in its deployment in the
form of the “‘tetractys,” was certainly presented as a triangular number
long before Aristotle. And what at first seems merely a game does
lead to arithmetical combinations that are by no means trivial. For
example, if the odd numbers, when added successively in a pebble
figure, make a square each time, this means discovery of οἰ rule for
the series of square numbers; and, if in the ἑτερομήκεις ἀριθμοί, construc-
ted in similar fashion from the even numbers, one recognizes the
triangular numbers, doubled, then he has the formula for the sum of
: eg
triangular numbers, a special case in the arithmetic series.
1 Above, ch. I 2, ἢ. 27. ψῆφοι provide the simplest tool for practical saga ει
the abacus (on which see M. Lang, Hesperia 26 [1957] 271-287). It ey e ae,
also that numbers have been indicated on dice, from very early pameael earl ΙΒ Νά
(see Daremberg-Saglio s.v. tessera). These figures, which make the difference be ui =
gain and loss, are naturally looked upon with a certain emotion and come to appes a
independent beings. In the substantive ἡ oivy is preserved the Indo-European ro
“one”: and the substantive ἡ μονάς was also used in dice games (Pollux 7.204). :
2 Met. to92bi1: οἱ τοὺς ἀριθμοὺς ἄγοντες εἰς TA σχήματα τρίγωνον καὶ τετράγωνον.
3 ve, ch. I 3, n. 120.
at ὅρι ΠΡ ΝΣ are numbers of the form n X (n + I) (see Theo ἊΣ iaag ee
Nicom. Ar. 1.19.19, 2-17.1, lam. In Nie. 74.19ff ). From the figure it can 45 8 =
r+2+3+4...+n=2(n+4 1) Χ n. In itself the word oe ee Ἶ
“having unequal sides” in general (cf. ἑτεραλκής, ἑτεροκλινής, ἑτεροπαχής, ἑτεροπλατής,
427
@eeodo
@e@eoo
@#ooo
οοοοο
> lhUh/ FF ee Θα Se ee BS δεν ΦῪ wr ee TIN é MIBK
— the ψῆφος diagrain can also be derived immediately the
, ) os :
ihe a of Pythagoras," which can be traced back as far as Anatolius
or calculating the rational sides of a right triangle, beginning with an
od a is |
d number;§ and this very fact has been rightly regarded as showing
that this is an element of ancient tradition. This of course presupposes
the knowledge of the Pythagorean theorem,” that most famous of all
features of Pythagorean mathematics, connected in the tradition th
the proverbial sacrifice of an ox. There has been much pees
over this tradition. Proclus, in his curiously twisted sentence a
hot in any case cite Eudemus.® The principal testimony ee of
two verses, cited several times but not known in their original context
from a certain Apollodorus, known as ὁ λογιστικός or ὁ ἀριθμητικό |
who may be the same as the Democritean from Cyzicus ae th :
case would be datable to the fourth century B.c.’ For him | alread th ;
γράμμα of Pythagoras and his offering of cattle are Cp aaeis ᾿ The £
that the offering of cattle featured in the story is in flagrant ae
a ς-Ἢὁ
τὰ Gia bie The restriction of its sense (cf. Iam. and Nicom. in the Passages cited
Seca as is pe case with τετράγωνον (above, ch. VI 1, n. 102), that the ee is tiie
see cs Se ἢ] 97). The general term for “ rectangular” in later times was προμήκης
perme. ee Be: (at Tht. 1476 it is used interchangeably oth
7s). Ῥ. $46c, Lim. 548; ἑτερομήκης in the 3
ce ate De an. 413a17, Cat. r1a10, Eucl. I, def. a ‘ ἐδ ἐμὰ ὦ
ee προ In Ἐμεὶϊ. 418.7ῆ. Hero Geom. 8 Heiberg IV p. 218; both are cited in the scholium
» P. 215.27; cf. 213.19ff. On the relation of the formula to the “on o
pig taney ) le see Heath, Math. 81; Heath, Eucl. 1 186. Berke: aD <:
on Fritz, AnnMath 1945, 252. The corres Ϊ ᾿ ; τὴ
᾿ | ν 252. ponding formula f ich i
ἘΠΕ to Plato, is hard to derive from the pebble aoe ee enna eeae
oe iia aan of the problem is Heath, Eucl. I 350ff. Proclus’ sentence (In
"ΤᾺ 4 οι : ape Tuns: τῶν μὲν ἱστορεῖν τὰ ἀρχαῖα βουλομένων ἀκούοντας τὸ
He ρήμα τοῦτο εἰς ἀλυθαγόραν ἀναπεμπόντων ἐστὶν εὑρεῖν καὶ βουθύτην λεγόντων αὐτὸ
τι τῇ εὑρέσει. The sentence 15 regarded as corrupt by DK and KR (no. 281); on ἀμὸν
τὴ δι ag object after ἔστιν εὑρεῖν. The scholium to Eucl. p. 213.1 sinpliae
ἊΝ Bier pica τὸ none τοῦτο εἰς Πυθαγόραν ἀναπέμπουσιν. Vogt ah nice
: : -—1909, I Ἵ Ϊ 5
Seated 9 ), has brought out emphatically that Proclus is not citing
vi τι», é
ἡνίκα ΠΠ|υθαγό ; ἐς εὖ 1 iv’ ἐφ᾽ 6
oo : Aa τὸ περικλεὲς εὕρετο γράμμα κεῖν᾽ ἐφ᾽ ὅτῳ κλεινὴν ἤγαγε βουθυσίην.
ee hte , 1.25 (ὁ ογιστικός), Ath. 10.418f (6 ἀριθμητικός;; in line 2, κλεινὸς ἐφ᾽ 5)
si Ν aga 1.1094b (ν.1. ᾿Απολλόδοτος; line 2, κεῖνος ἐφ᾽ ᾧ λαμπρὴν ἤγετο) Probably
“Ὁ, AP. 7.119; RE s.v. Apollodoros (68). On Apollodo das,
" , 8). On Apo odorus of Cyzicus, see ab
ae ; "ἢ ΒΕ. ne ths aan had ἐμὰ Νὴ his school in Cyzicus, and his pupils Helicon
: me from there; but τῆς Democritean Bj f
also a mathematical astronomer, so th j τὸ push eine Ade
: at the Democritean Apollodorus mi
eee ! night wel
ie ss ἐπε ν The verses do not say which “famous” figure of tela es a
] n, but D.L. and Ath. call it “the Pythagorean theorem,” and Plutarch presupposes
428
1. Pythagorean Arithmetic
tion with Pythagorean vegetarianism ought rather to be considered
an indication of antiquity than the reverse.®
What Neugebauer first suggested as a possibility in 1y28 has since
then grown into a certainty—namely, that the “ Pythagorean theorem”
had been used routinely for centuries in Babylon, and was therefore
obviously not a discovery of the Greeks.® It must have been introduced
as a piece of Babylonian arithmetical technique. It is possible that
Pythagoras was the intermediary; but, in view of the multiplicity of
contacts between Greece and the Orient in the sixth century, the
“fame” of Pythagoras can hardly be explained on this one ground.
There is no testimony that he gave a strict proof of the theorem, and
this cannot be made to seem probable!° The suspicion remains that
the theorem had more than a mathematical significance in Pythagoras’
school, and that the numbers involved seemed in a cryptic way
meaningful. The formula of “Pythagoras” points in this direction,
as it belongs to the context of the pebble figures, like the form of the
tradition that only mentions the triangle with the sides 3, 4, and 5.
In fact, this fits especially well with the kind of number speculation
we learn of from Aristotle, where 3 is male, 4 is female, and 5, which
mysteriously unites them in the Pythagorean triangle, is “ marriage. ae
Plato’s “nuptial number” obviously presupposes this interpretation.”®
What we are considering here, then, is not Pythagorean geometry, but
arithmetic, developed by speculative interpretation of Babylonian
formulas.
8 Above, ch. II 4, n. 110. Cic. Nat. 4. 3.88 is dubious about the sacrifice of an ox; he
passes over the geometrical problem with the neutral phrase “in geometria quiddam.” -
One cannot simply brand the tradition “impossible,” as van der Waerden docs (SA
100).—Pamphila, ap. D.L. 1.24, seems to be transferring the Pythagoras story to Thales,
® Neugebauer, NGG, math.-ph. KI., 1928, 46-48; QSt 4 (1936) 247; EwSe 39) ef.
Becker, MD 10, 55; van der Waerden, SA 76ff; Stapleton, Osiris το 48, 12ff A table
(“Plimpton 322”) with “Pythagorean numbers”’ was published by ©. Neugebauer and
A. Sachs, Mathematical Cuneiform Texts (1945) 38-41. The “Pythagorean theorem”
was also known in India (Apastamba~Sulva~Sutra; cf, Heath, με | 3att, Math, 1 14 tf,
Becker, MD 55ff)—though Greek influence is not impossible here sand in’ China
(Becker, MD 56; cf. below ch. VI 4, n. 38).
10 The classic “windmill proof” comes from Euclid, but more primitive proofs are
possible; see Heath, Eucl. I 352ff, Math. 1 147ff; Becker, MD 446} It is pure guesswork
to suggest that Pythagoreans tried anything of the sort.
11 Vitr. 9 praef. (mentioning the sacrifice of an ox), Nicom. ap. ‘Th. ar. s0.21ff fam.
VP 130f, cf. 179 (and Delatte, Pol. soff), Alex. Met. 75.27ff (from Aristotle?).
12 Above, ch. I 2, nn. 26, 62; cf. also Plut. De Is. ef Os. 56 (3 is Osiris, 4 is Isis, and ς
is Horus), Greg. Naz. Ep. 198, Philo Vit. cont. 65, Schol.D. Th. tg.119.
13 ἐπίτριτος πυθμὴν πεμπάδι auluyeis, Rep. 546ς; brought into connection with the
“Pythagorean triangle” in Iamblichus, Alexander (above, n. 11), Aristid. Quint. 3
p. 151f M. Cf. also below, ch. VI 4.
429
y= & EEE ΑΔ ΔΕῪ EVASION LFTB VY
Based on the “Pythagorean theorem,” there appear the Pythago-
reans’ “side numbers" and “diagonal numbers," a series that provides
increasingly accurate approximations to the value of J/2. Plato
apparently knew about this calculation. The manner of constructing
the serics can be perceived from a simple jigsaw-puzzle procedure;
and the theoretical basis for the procedure is given in the tradition in
the form, “The unit, as the origin of all things, is both the side and the
diagonal of the square.’’!7
concept of irrationality.18
Further reconstruction of Pythagorean arithmetic depends on the
question, to what extent the detailed expositions in Theo, Nicomachus,
and Iamblichus may be regarded as evidence for early Pythagorean
mathematics. They are usually treated, in modern accounts, as ‘‘ Pythag-
orean arithmetic,” before Hippocrates of Chios,!® and no attempt is
made to separate more ancient and more recent material. It is inad-
missible, however, to avoid the question of the form of this “ Pythag-
orean arithmetic,” for this is the crucial question in a consideration of
its scientific character. The compilations from later antiquity offer a
goodly number of arithmetical facts, but without any
general proof of any of them. The rule is illustrated by a few examples,
and this passes as verification. Of course, such incomplete inductive
procedure leads to errors.2° The books of Euclid which are devoted to
arithmetical matters are quite different: general and rigorous proof,
more geometrico, is the rule, the numbers being represented by line
segments. Is the inductive, proofless form ancient and primitive,?!
This does not in any way presuppose a
attempt at
14 Theo Sm. 42.10ff, Procl. In Remp. II 24.16ff; cf. Heath, Math. I ΟἹ, Eucl. I 308ff;
van der Waerden, SA 126f;: Becker, MD 67f, 73f.
15 The number 7 is διάμετρος ῥητὸς πεμπάδος (i.e. 4/50): Rep. $46c.
*° Side plus diagonal gives a new side: 2 sides plus a diagonal give a new diagonal.
Van der Waerden gives a geometrical proof (SA 127), but the adjoining figure is still
more perspicuous: from 3 isosceles right triangles a new one is made.
Ὁ Theo Sm. 43. 5ff.
18 διὸ καὶ of Πυθαγόρειοι ἐθάρρησαν τῇ μεθόδῳ, Proclus writes—(In Remp. II 25.9ff),
because the inaccuracies cancel each other out—the approximations to 4/2 are alternately
too great and too small. This does not sound as though the impossibility, in principle,
of attaining an exact result, was seen as important. In any case, this method can be dis-
covered by induction, without depending on Euclid 2.10, which is given in the late
sources.
19 Heath, Math. 65-117; van der Waerden, SA 92-127; Becker, MD 4off: cf. von
Fritz, Gnomon 30 (1958) 82ff; van der Waerden, RE XXIV 280-28 5.
“0 For example, the conjecture is erroneous (lam. In Nic. 33.
number in each category (units, tens, hundreds, thousands, etc.), and that they ended
alternately in 6 or 8 (Nicom. Ar. 1.16.3). They only worked out the first 4 perfect
numbers (6, 28, 406, 8128); the next turns out to be 33,550,336 (Heath, Math. I 74f).
*? As van der Waerden thinks, SA 97, RE XXIV 283f.
20) that there was 1 perfect
430
or is it the prevlis tak cer adenee aA dilute, popularizing selection from
what had been a rigorous mathematical system a | ᾿
There did exist a continuous arithmetical tradition which by passe
Euclid. Aristotle knows of the ψῆφοι numbers, which reappear in
Nicomachus and Theo, whereas Euclid represents the nunibers as
lines; Aristotle speaks of triangular numbers,?* Euclid only of srs |
(that is, rectangular) and “solid” numbers. Certain intermediate
stages of this extra-Euclidean tradition can be aoe coe
with Plato’s immediate pupils: Speusippus wrote Περὶ ees
dpiOudév—but he also included non-Pythagorean material; +
crates, too, wrote books entitled Τῶν περὶ ra μαθήματα βιβλία τι
Περὶ ἀριθμῶν, ᾿Αριθμῶν θεωρία (D. L. 4.13); a Philip of aes ules
᾿Αριθμητικά, Περὶ πολυγώνων ἀριθμῶν, Κυκλιακά, Μεσότητας."" Later on,
Hypsicles too wrote on polygonal numbers.” | -
Certain advances beyond the pre-Aristotelian stage are clearly
discernible: Aristotle speaks only of triangular and square cares
and the expansion to polygonal numbers probably does not pes
Philip of Opus. It is more important that there is no aie - δὴ
Aristotle, of ‘“‘perfect’” numbers in the sense of Euclid. | For t :
Pythagoreans, as Aristotle knew them, the number 10 is per
neither 6 nor 28 plays any role. Nor does 28 occupy any signi τ
position in the later tradition.?* Euclid’s definition of “ perfect “a @
is too abstract to catch the imagination of the lovers of symbo ism. It
is foreshadowed in Plato’s Laws, where the number of citizens is 2
be 5,040 because this is a number with a particularly large number o
22 So Heath, Math. I o8f; Capparelli II 451 (with a ref. to the fact that Euclid too was
simplified in the Middle Ages).
23 Above, ἢ. 2.—Eucl. 7 defs. 17-18.
24 Speusippus fr. 4; cf. above, ch. I 3, n. 106.
25 Suda s.v. φιλόσοφος.
26 Diophantus ed. Tannery I 470.27, 472.20. ee .
27 reneran number is one which is equal to the sum of its divisors, including 1; τῆν
6 is divisible by 1, 2, and 3, and 1 + 2 - 3 = 6; and 28 is divisible by ee ee =
ae i in detail by Theo Sm. 45.9ff, Nicom. Ar.
ich add up to 28. The matter is treated in
= ea In Nic. 32.20ff; cf. Heath, Math. I 74ff; Becker MD 47ff; van der Waerden,
lso including “friendly numbers”’); above, n.20. .
geek ἀεῖτοι με ἘΠ 28 is mentioned in Diocles and Strato (frr. 97-98 ὙΠ:
Nicom co Th. ar. §9.14ff, 66.13, Philo Op. ror (from ee neces ue Ῥ'
(Es he 28 tones of the tone system with reference to the Ti y
275.16 (cf. 280f) interprets t : ἢ ἐλ θα οι τ τῳ μις
‘ ea tween the two ideas of a “p
not to the “ perfect number.” The difference be a ctuneaaern
ἃ Theo Sm. 46.12ff—The Pythag
is noted by Nicomachus ap. Th. ar. 44.15 an eae ae
in A. P. 14.1 gives 28 pupils of Pythagoras, just as, according : :
a 28 ice Tannery emphasized that the Euclidean perfect number was not the
same as the early Pythagorean (HScH 382).
431
divisors," and in the context of the “generation” of numbers in
Plato's system of derivation, it was inevitable that the question sf the
divisibility of larger numbers come to the fore. But the Euclidean
perfect number” is not a feature of carly Pythagoreanism.
There are variations in the terminology, too. For Plato and Aristotle
2 is an even number,®° for Aristotle and Euclid a prime number;?!
but in the later ““Pythagorcan” sources it will not fit the definition of
an even number* or that of a prime8—in fact it is not regarded as a
number at all, any more than τ, but constitutes, along with the latter
the first principle of number.** This is sometimes regarded as ees
because it is illogical ;35 but it might better be regarded as a develo mien
from the Platonic system of derivation, in which the aoe have
their origin in & and δυάς.36 On the other hand, the later sources show
a more developed logic in the subdivision of even numbers; while in
(τρία δίς) are defined side by side, the later writers distinguish ἀρτιοπέρισ-
σος from περισσάρτιος in such a way that the two concepts are egal
exclusive.>* The conception of the ἀρτιοπέριττον is quite different
in the Pythagoreans; here it is one that has a part in both terms.3® As
early as Aristotle a change is evident to the more developed ἘΠ ἫΝ
matical terminology; in a passage where he is “Pythagorizing” on
his own, he calls the number 6 aprtomrépirrov.*°
Ἐν far as concerns the method employing deduction and proof,
the later tradition shows, in one respect, a decline, giving Euclid’s
29
pena: 7:1 et seq. τὰ Aaa 5,040 is divisible by all numbers from 1 to 12
: eaning o Is τέ
ete & Of ἀριθμος τέλειος in Rep. 546 is a moot question (cf. also ch.
= Pl. Parm. 1436, Arist. Top. 157439.
1 Arist. Top. 157439, Eucl. 7 def. 12.
32 Cf. Nicom. Ar. 1.7.3f.
Se KY:
a eee ey . Ξ : oe Jam. In Ole : 8ff present the prime numbers as a subdivision
a m. 23.14ff calls them περισσά ΐ i
special position, is also called περισσοειδής es a cee
34 Nicom. ap. Th. ar. 9.1ff.
36 Heath, Math. 1 71; Becker, MD 46.
= ae i ch. I 1, I 3, and esp. Theo Sm. 20. 5ff.
. Parm. 143e, Eucl. 7 defs. 8-11. Because of the i
5, : mathematical equivalence of th
two Pass some have athetized def. 10 (cf. the controversy as “μὴν as Iam In Nic.
pete ) In any case, the agreement with Plato is noteworthy. . “
δι ai ig le 1, Theo Sm. 23ff, Iam. In Nic. 20.14ff: ἀρτιοπέρισσος is a number
Mass ie τὰ ios x (2n + 1); περισσάρτιος is one whose half is in turn even, 2™ x
; ); ucli inds it necessary to express this in a clumsy circumlocution: ori
TE ἄρτιος καὶ ἀρτιάκις περισσός (9.34). ΠΣ
39 Above, ch. I 2, n. 39. On Philol
ove, ch. I 2, n. 39. aus fr. 5, see above, ch. III
40 Arist. fr. 47; above, ch. I 3, m. I6I. eee
432
oc Pere rer eee
cule for “perfect numbers” without Euclid’s proof, but, on the other
hand, Speusippus obviously proceeded in no less inductive a manner
than his successors. Hlypsicles was a professional mathematician, but
some of his propositions were not furnished with proofs before
Diophantus.*? The inductive, proofless kind of arithmetic, which
lasted till long after Plato, may be regarded, along with a few basic
rules, as belonging to the carly Pythagoreans.
This is confirmed by the amazingly close agreement of this inductive
arithmetic with Babylonian techniques of calculation. There too, by use
of rules which are unproven, series are developed and made available
for practical use in tabular form: series of squares, of cubes, and even of
“Pythagorean” numbers. In fact, when so unusual a series as that of the
form n? (n + 1) makes its appearance, with the name παραμηκεπίπεδοι,
in the Pythagorean tradition, we cannot but think of direct influence.
It is possible that this Babylonian import was introduced by Pythagoras
himself; but it was altered in the process, at least insofar as, in Aristo-
xenus’ words, the numbers were withdrawn from the use of merchants
and “honored” for themselves.“# The graphic procedure with ψῆφοι
makes it possible to formulate impressively generalizations about
numbers; but it also “‘reveals” each fact without deducing one from
the other in an abstract chain of reasoning. It is the element of the
unforeseeable which gives number games their appearance of some-
thing profound and secret. The “occult” charm of mathematics comes
from the fact that the human mind forgets its own way of proceeding
and loses sight of its own preconceptions; for alert mathematical
analysis, that which fills the naive mind with amazement 1s secn as
tautologous and therefore self-evident. What we find among the
Pythagoreans is amazement and “reverence” for certain numbers and
their properties and interrelations. “Even” and “odd” are united in
“marriage”; and to them this means that cosmic forces are at work. A
scheme of proof could hardly be anything but annoying because it
would show the result as the logical consequence of the preconceptions,
and reduce it to banality.
Even a game may be regarded legitimately asa kind of mathematics ;48
41 Stressed by Tannery, HScH 387.
42 Diophantus ed. Tannery, I 470.27 (cf. 472.20): ἀπεδείχθη τὸ mapa. “ὙὙψικλεῖ ἐν ὅρῳ
λεγόμενον.
48 ©. Neugebauer, Οὐδὲ 4 (1938) 181-192, on Nicom. Ar. 2.16.3. Cf. above, ἢ. 9;
van der Waerden, SA 45.
44 Fr, 23; above, ch. VI 1, ἢ. 77.
45, Lorenz, Arithmetik und Logik als Spiele, Diss. Kiel, 1961.
433
"et S&S ee ee δι SEY ἐγ EVE EF ETEN ὃ SAE AINE
the axiomatic-deductive forny is not the only one possible. But the
distinctive achievement of Gireck gcometry is that, for the first time, it
created such an axiomatic and deductive system. Seen in the context
of this Greek geometry, Pythagorean arithmetic is an intrusive,
quasi-primitive clement,
By analysis of Euclid, Oskar Becker has reconstructed a set of
theorems which has been widely heralded as proof of the existence of
a deductive Pythagorean arithmetic,4® namely the “doctrine of odd
and even,” as developed in Euclid 9.21-34. These propositions stand
isolated in Euclid, a trivial appendage to the sophisticated number
theory of books 7-9, which culminates in the proof that there are
infinitely many prime numbers (9.20). The propositions about the even
and the odd are only once applied by Euclid, in the proposition about
perfect numbers (9.36), and in a proof of irrationality, given as an
appendix, which Aristotle already knew.4? On the other hand, the
basic importance of the even-odd antithesis for Pythagorean cosmology
is well known; and since a fragment of Epicharmus uses the antithesis
of odd and even, the conclusion has been drawn that the Pythagorean
odd-even doctrine was alrcady so well developed and so widely
known about 500 B.c. that it would be recognized in a comic allusion.*®
The fact that most of the propositions can be proved with simple
pebble figures gives an important piece of confirmatory evidence; so
the “doctrine of odd and even” is regarded as the best-attested element
of very early Pythagorean science, in the form of deductive mathe-
matics.
Yet this structure, apparently so firm, has fissures that must lead
inevitably to its collapse. It is true that the Pythagoreans were concerned
with even and odd numbers; but this is far from proving that they
sct forth the propositions found in Euclid, in sober mathematical
formulation, and provided them with proofs, or that the propositions
6 Becker, QSt 3, 533ff, repeated Grdl. 37ff, MD 47ff. Agreeing, among others, van
der Waerden, MtAnn 1949, 127ff, SA 108f; Reidemeister 31ff; Szabé (above, ch. VI 1,
n. 135); von Fritz, RE XXIV 202.
11 Eucl. 10 App. 27 Heiberg; Arist. Av. pr. 41226, 50437. It is proved that 4/2 would
have to be even and odd at the same time, and is therefore not expressible in numbers.
Cf. Heath, Math. I 91, Eucl. ΠῚ 1; Burnet, EGP 105; van der Waerden, SA 110; Becker,
MD “τὸ. Mondolfo, Inf. 241, asserts simply that the proof had been “‘attribuita da
Aristotele all’antico pitagorismo”; but this is baseless (cf. n. 109). Becker, QSt 3, $47,
finds a hint of this proof in Pl. Leg. 819d, “‘if these words have a precise sense” (meaning
a precise mathematical sense). Cf. also below, ἢ. rog.
48 Epicharm. fr. 2, cited by Becker QSt 3, 537; Mondolfo in ΖΜ 318ff; von Fritz,
RE XXIV 204.
434
about irrationality and perfect numbers belonged with them from the
beginning.
Becker himself assumes that Euclid reworked the Pythagorean
doctrine, and reconstructs simpler proofs with pebble figures, But this
transformation has deep consequences. In Euclid, the theory unrolls in
systematic fashion, and onc proposition presupposes the preceding, in
the strictly deductive manner, whereas a proof with ψῆφοι is essentially
inductive and pictorial. According to the principle of perfect induction,
it can be regarded as probatory, but it does not presuppose other
propositions; every set of facts is evident in itself. There is no need
for a systematic structure, which is of the essence of deductive mathe-
matics. Of course the Pythagoreans knew that odd plus odd makes
even, and that odd plus even gives odd—they demonstrated this with
their pebble diagrams**—but they did not deduce one proposition
from the other. They saw, directly, that the “male’’ odd number
showed itself dominant in association with the “‘female’’ even number.
The propositions about perfect numbers and about the irrational are
on a quite different level. To be sure, Becker can derive from a pebble
figure an important lemma for the proposition about the Euclidean
perfect numbers,°° but to go further than this requires an abundant use
of modern algebraic notation. Here is presupposed a whole chain of
logical conclusions based on very precise concepts of divisibility,
prime numbers, and factors, which is totally inconceivable without a
written system of στοιχεῖα and a quasi-algebraic method of representing
numbers by general symbols.*! In addition, it 15 probable on other
grounds that the Euclidean idea of a “perfect number” was first
developed in the Academy (above, nn. 27-28). It is not part of ancient
Pythagorean speculation about the odd and even.
49 Becker imagines that they represented even numbers by equal numbers of white and
black pebbles; but an arrangement in two rows is even more striking (cf. Pl. Euthyphro
12d, where even number is defined as ἐσοσκελής and odd as σκαληνός):
ecce ee _e@ec0ece
eee tee Γφοοοο
Euclid 9.21: even +- even = even re ee _eevscee
Euclid 9.22: odd + odd = even eco tTeoe evecce
even + odd = odd eee Coo eeeeece
eco teee “eeceoce
(But see also above, ch. I 2, n. 31. The “power” of the odd number depends on the
“one” that it has in its ‘‘ middle.”’)—In Euclid, 9.22 presupposes 9.21. ;
50 Namely, the summation formula 14+ 1+2+4+ 8... Ἐ25 -- antl (QSt 3,
539ff). . . :
51 Archytas, like Euclid, represents numbers by line segments (Ά 10).
435
ΠΡ ἫΝ ὙἍ ΣῪ EVERETT ELAN ἢ κα τ καὶ τ
The same sort of thing ws πὸ for the proof of irrationality, Here
too, Becker tries τὸ reconstruct a proof with pebble figures :52 “Given
ee = 268. Think of the number αὐ laid out with pebbles once
and δ. twice..." Now, there is no question of the Pythagoreans
merely “thinking of" their ψῆφοι; they held them in their hands and
arranged them in visible patterns, as is obvious from the testimonia
about Eurytus.®? If one tries, however, following the spirit of Becker’s
reconstruction, to represent the numbers a2 and b? with counters, one
quickly realizes that this is impossible; there are no whole numbers
which will exemplify the formula a2 = 2b2. Becker demands, however,
that the Pythagorcans proceed, undaunted by this experience, to
“think of” the unrealizable condition as met, so as to go on drawing
further conclusions from this contrary-to-fact postulate, until the logical
contradiction springs to view (b would have to be even and odd at the
same time)—the contradiction which “ proves” the absurdity of the
merely “thought of” postulate, which from the beginning flouted
common sense. Such a train of thought has nothing in common an
more with pebble figures and is compelling or meaningful only where
the demands of strict mathematical method are already recognized;
simpler souls will be satisfied with the inductive discovery that there
are no pebble diagrams to be devised which satisfy the equation
a” = 2b*. The arithmetic of ψῆφοι and the theory of the irrational are
mutually exclusive.®4 What is more, we learn from Aristotle that the
Pythagoreans considered 1 as both even and odd, and from Theo of
Smyrna that the Pythagoreans employed the unit as both the side and
the diagonal of a square.®> The axiom basic to the arithmetical proof of
irrationality, that every number is either even or odd, tertium non
datur, simply does not exist, in this form, for Pythagorean speculation.
This gives us all the more reason to suspect that irrationality had
originally no part in it and was rather discovered in the field of geom-
ctry, where the existence of the irrational js immediately obvious.%
δ MD 51; cf. QSt 3, $44f.
3 Above, ch. I 2, n. 69.
"Ὁ Reidemeister (52) says of the irrational: “It cannot be illustrated, only thought and
inferred.” Becker’s illustrations, consisting of empty squares, are quite different from
other pebble diagrams (cf. MD 52 with 4off).
55 Above, ch. I 2, n. 39; VI 2, n. 17.
58 Von Fritz, too (AnnMath 1945, 25 5), thinks that the irrational was first discovered
in geometry; see below, ch. VI 3. For Plato, the expression ἄλογοι ὥσπερ γραμμαί is
already proverbial (Rep. 534d); and ἀσύμμετρος is also a term coming from geometry.
If it is ἄρρητος (Rep. 5460), “not expressible (in numbers),”’ this presupposes the existence
of the corresponding object. True to its ontological basis, Greek mathematics, unlike
the modern, only develops concepts for “existing” magnitudes.
436
2. Pythagorean Arithmetic
Only the propositions about the perfect numbers and wees
irrationality give mathematical substance to the “doctrine ὁ ᾿ ι : |
even”; without them all we have isa remnant of very ieee) ἐν
disconnected theorems. Becker himself believes that Euclid ree —
proofs; but how simple it would have been for ἃ pre-Huc aa
mathematician to put the propositions about the odd and oS on ;
systematic paradigm of deduction! The proof of τ γε μὸ neg :
proposition on perfect numbers presuppose this dager ane
do not belong to the oldest stratum of Pythagorean arit ae he
“doctrine of odd and even” only later became part of deductive
mathematics, influenced by the mos geometricus.
»---
[ There remains the division of numbers into odd and even, -puieth
in which one might, in spite of all, see the ee — οἰ
theory; it is confidently attributed to Pythagoras aes Ps — .
very point can be refuted by philological ἐν rs all Py = "Ἢ
speculation the odd number is more highly valued; it — *
the limit,” is the male element, and stands in the column of t : B00
In all this is preserved, as shown both by anthropological para ᾿ rom
folklore and linguistic observation, an ancient and widesprea hte
of number lore.*8{ But in the Greek language the ve eas is
“well-structured,” ἄρτιος, and the odd is “excessive, περιττός in
normal usage ἄρτιος is unequivocally the one with wiser ee
tion, and περιττός is negative, representing a transgression © t enorm.
Thus Greek terminology for even and odd is in its tendency nein
ally opposite to the Pythagoreans’ numerical theory. oar ar iene
is easily comprehensible from everyday use: dividing τ 2 fee
time immemorial had an important role in practical ife, an ν "
no surprise that the words for “μα ἢ are built on a special root, Ww "
is not true of either “third” or “‘fourth.’’®° In Greek lands it was o
necessary to match two semi-choruses in a ritual eal or to ΠῚ
pairs in an agon—in Plato’s view this is the sort of example upon w
. ᾽, » O
57 Heath, Math. 1 70: “The distinction between odd (περισσός) and las — s)
doubtless pes back to Pythagoras.” Rostagni, Verbo 27: a gee ae Ὧν spa i
igine pi ica” might wo
ienti i d’origine pitagorica” (though one
scientifico, tutti sanno essere gC oe
“scientific value” in the Pythagorean tradition). See also Sarton, Hist. 204
58 h. I 2; below, ch. VI 4. |
59 ΠΣ see Solon 3.39 Diehl, Theognis 154, 946; on περισσός, Empedocles fr
is signi : σὺ Kal τὰ L ἐπίστασαι ἄρτια
13. Line 18 of Cleanthes’ hymn to Zeus is significant: σὺ καὶ τὰ περισσὰ ἐπὶσ p
θεῖναι. ;
' he τ
80 ἥμισυς is Indo-European in root and meaning.
437
Vie PYTAHAGORSAN NUMBER THEORY
children should practice calculation”! Either the procedure is successful,
because the number is “symmetrically constructed,” or one person
is left over—reperrevier.™ [t did not need a Pythagoras to produce the
insight that the problem “comes out right” with 2, 4, 6, 8, or 10
persons, but not with 3, 4, 7, ory. Just asin Latin, English, and German,®
the Greck distinction between even and odd grew out of elementary,
everyday calculations; but then, among the Pythagoreans, it took ona
different and unpredictable significance.
The term Adyos, in its mathematical sense of “relation, ratio, pro-
portion,” has been attributed by von Fritz to the Pythagoreans, and,
conjecturally, to Pythagoras himself.® Its origin lies, he thinks, in
Pythagorean musical theory: the λόγος, a “Mitteiling von etwas
Wesentlichem an einem Gegenstand,” “the communication of some-
thing essential about a thing” (81), is, in relation to music, the numerical
ratio present. If one knows this, he knows the nature (Wesen) of the
interval, and can reproduce it.6* And similar reasoning holds in the
realm of geometry; the ratio 3:4:5 determines the shape of a right
61 Pl. Leg. 819b.
* Cf. Hes. fr. 278 M.—W. (from the Melampodia): μύριοί εἰσιν ἀριθμόν, ἀτὰρ μέτρον
γε μέδιμνος, εἷς δὲ περισσεύει (i.e. the bushel measure holds 9,999 figs).
*8 Neither par and impar nor gerade and ungerade are borrowed from Greek. English
odd comes from the Old Norse, and originally designates the third or other “odd”
man in a council, whose vote decides in case of a tie (NED VII $8ff).
84 Epicharmus plays with the ideas of odd and even (DK 2382 = fr. 170.7ff Kaibel =
Alcimus FGrHist 560F6 = D.L. 3.11; cf. above, ch. III 3, n. 58; the authenticity of the
fragment was denied by Wilamowitz, among others, Platon II 28.2): <ai> πὸτ ἀριθμόν
τις περισσόν, ai δὲ λῇς, πὸτ ἄρτιον ποτθέμεν AR ψᾶφον ἢ καὶ τᾶν ὑπαρχουσᾶν λαβεῖν, ἢ
δοκεῖ κά τοί γ᾽ «ἔθ᾽» ωὑτὸς εἶμεν; ... Luckily, in this case we know the comic context
in which the words occurred. A debtor is trying to free himself of his debts by using
the claim that he is no longer the same person; we know from Chrysippus (ap. Plut.
Comm. not. 1083a; cf. Pl. Tht. 152d-e) that this περὶ αὐξήσεως λόγος did occur in Epi-
charmus. When something is added or subtracted, what is left is no longer the same thing;
all of us are constantly gaining weight . . . and so on. The proposition is proved from two
examples, involving number and size respectively. This restriction is not specifically
Pythagorean, but essential to the argument, which is only valid with relation to quantity.
The further subdivision of numbers into odd and even seems less essential (Rostagni,
Verbo 28f), and therefore an indication of external, presumably Pythagorean, influence;
but, for the idea to be clear enough for the comic stage, it was necessary that the general
concepts be replaced by something more specific. This is why the idea of measure is
expressed from the start as παχυαῖον and the idea of number as even/odd. By the addition
of a single pebble a number changes from odd to even or the reverse; it is precisely
the notion of odd and even which makes it clear that the very smallest alteration signifies
a fundamental change. Even if one were to assume that there is a relation between this
passage and the Pythagoreans, the “‘number theory” presupposed would be the inductive
and graphic, rather than the deductive. See also von Fritz, RE XXIV 203-205.
85 ABG 1955, 81ff (cf. RE XXIV 199).
66 Though not in a practical way; cf. Aristotle, cited above ch. V 1, n. 62.
438
triangle, and with ats help one can reproduce the right Ἀμμ], Thus the
λόγος would be “the group or bundle of numbers that lie hidden tn al
thing, by use of which it can be not only described but reproduced
(83). |
Now a distinctive feature of the Grecks’ calculation of proportions,
and in particular of Pythagorean musical theory, is the occurrence of
certain terms like ἐπίτριτος λόγος, ἐπόγδοος λόγος, and the more general
ἐπιμόριος λόγος, which from a German or an English point of view
seem odd. Their sense is quickly evident; ἐπίτριτον, for example,
“a third in addition,” means 1 + 4 or ἃ, But the question remains,
what made these ratios so important that they alone, in the Greck
language, have special names, while a fraction like 3, unlike 8 (ἐπί-
πεμπτον), can only be expressed in a cumbersome circumlocution.
The answer is simple: they were terms used in the calculation of
interest. Whoever lends money expects to get his principal back and
a specified fraction of it in addition. This could be ἐπίτριτον (3, or
334%), 87 ἐπίπεμπτον (δ, Or 20 °/),68 more usually ἔφεκτον (6, Or 1627)"
sometimes ἐπόγδοον (2, or 123 Ὁ eat te lowest ἐπιδέκατον τόκον}
(interest of +5; this was what the gods received).” It is certain that the
practice of loaning money at interest went back before the time of
Solon; and, though there is no direct evidence, it can hardly be
doubted that the expressions mentioned were in use that early—long
before the day of Pythagoras.”* Thus when terms like this turn up in
a musicological context—énizpirov and ἐπόγδοον in Philolaus (fr. 6)
and the general ἐπιμόριος λόγος in Archytas (A19)—they are borrowed
from everyday speech.
But the calculation of interest is in fact called λογίζεσθαι : λογίσωμαι
87 Xen. Vect. 3.9, Isaeus fr. 79 Sauppe, Arist. Rhet. 1411a17.
68 Xen. Vect. 3.9.
69 Demosth. 34.23.
τὸ Demosth. $0.17. |
71 IG [2 377.12 (434/432 B.c.); cf. IG VII 4263 = SIG* 544 (Oropus, ard century B.C.):
whoever lends the city money at this rate becomes πρόξενος Kat εὐεργέτης; IG IX
61a79, Arist. Rhet. 1411a17, Oec. 1346b32.
aa 1. θεοῦ τὸ ery ae IG 15 45.25 (446/441 B.C.), 39.35 (446 B.C.), Xen. Hell.
1.7.10 et saep. ἐπιδέκατον means here simply “a tenth,” as ἐπίπεμπτον means “4 fifth ᾿
(Ar. fr. 201 = Harpocr. s.v. ἐπίπεμπτον, with further refs. ; also IG va 3073.1 = SIG
972.1); but it is precisely the tenth or fifth that must be added, extra, because the other
party lays claim to it. In the Sagras battle, the Locrians outbid the Crotoniates by vowing
a ninth—évarevew instead of Sexarever (Justin 20.3.3). ΝΝ
73 That proportion was used at this time is also shown by the expression ἐκτημόροι
(on which see Ὁ. Lotze, ‘““Hektemoroi und vorsolonisches Schuldrecht,” Philologus
102 [1958] 1-12).
439
ΚΠ ΔΟΘΝΚΒΑ͂Ν NUMBER THEORY
τοὺς τόκους, says Strepsiades. ‘The officials who calculate the interest
= ne a are called Aoyrad,?® and the calculation itself ᾿
eyes. Heraclitus writes: “the turnings of fire:
half is earth, half burner . .. (Karth) is ane as evita 4
ιῦ the same proportion as it was before it became carth” (kai per, ae
εἰς τὸν αὐτὸν λόγον ὁκοῖος πρόσθεν ἦν ἢ γενέσθαι γῆν). 77 Here i coy -
eee sense of the word is clearly present. Later one said κατὰ ae
αὐτὸν Adyov,”® or simply ἀνὰ λόγον “in the (ascending) sequence of
calculation.” Thus ἐπίτριτος λόγος means “calculating at ey 7
λόγον διδόναι means “render account,” and ὧδ᾽ ἔχει λόγος ; “hi : ‘he
state of the account.” The sense “calculation” comes from the δ:
sense of the root Aey- almost more directly than the sense “word "79
But Pythagoras has nothing to do with all this. Ν
τ connection of proportion and music, resulting in the equation
of διάστημα with λόγος, remains to the credit of the Pytha ereans:
and in one aspect of the theory of proportion, the doctrine of τ ”
Pythagorean influence is a possibility. The three μεσότητες, the ahs
metic, gcometric, and harmonic means, are generally regarded in the
tradition as a discovery of Pythagoras.®° The fact that all three reat
have a role in the Timaeus® could rouse suspicions about the tradition
involving Pythagoras himself. But Theaetetus already knew th
system of the three means, and used it, ina rather forced ae as ὡς
point of departure for his classification of irrational lines ;82 aa the
74 /
: ΠῚ pe ates a to the elementary techniques of calculation learned
: ' mination an j 1
ieee most difficult part was the calculation of interest
76 IG I? 324.1, 72 (426/422 B.C.).
8 Pi. OL. 10.11.
™DK 221331. The su ἢ ]
' pplement «γῆ», Ι
wee vq, rejected by Reinhardt (Hermes 77 [1942] 16),
78 IC) 12 . yA n
ee ; ᾿ ἡ: (423/422 BC) Kata λόγον τῆς τεταγμένης ἀποφορῆς, Hdt. 2.109; λόγον
i ἂν dt. 3.142f; IG I 91.25 (decree of Callias). Cf Hde. 1.134, 2.13, I 68 a
ie Fi 4.451: λέκτο ὃ ἀριθμόν. Cf. esp. καταλέγειν, κατάλογος. δυὸ
“ τι Sim. 113ff, Nicom. Ar. 2.21ff, Iam. In Nic. 100 15ff.
. « geometric mean as δεσμός, 31¢c-d: ican
ik oe Hos, 31c-d; the arithmetic and harmonic means, 36a,
"This is attested by E
$4 y Eudemus in a fragment preserved only i
, ἘΠῚ ΠΆΠΑ
oa Ho by Wehrli): Pappus, Comm. on Euclid X, ed. G. iia and eee
Peat ae ἐστι Σ Woepcke, Paris, 1855) 1.1 p. 63: “it was peventcles
Heretus who distinguished the powers (i.e. the ich a
tiie ὲ ὑ νον: (δ. squares) which are commensurable in
; € incommensurable (i.c. in length) [this is fi 2
147d ct seq.] and who divided the more a i aba Ta
> gencrally known irrational lines accordin
» *. + 1 Η 1 :
different means, assigning the medial line to geometry, the binomial to sane a
440
2. Pythagorean Arithmetic
means are prestunably older, and they are closely related to Pythagorean
music theory. ‘The name of the “harmonic mean”’ is to be explained
directly from the latter; the Mese is the harmonic mean of the octave
Nete-Hypate.®* In fact, it is attested that Archytas and Hlippasus
introduced the term ἁρμονικὴ μεσότης in place of the older ὑπεναντία
μεσότης and the series 6, 8, 9, 12 is presupposed in the experiment
which Aristoxenus ascribes to Hippasus.®> If this third and most
complicated of the means was known, then the two others surely were."
It would be conceivable that the harmonic mean was discovered in the
context of Pythagorean music theory;®’ but it also has another use:
along with the arithmetic mean it provides increasingly closer approxi-
mations of the square root;®* and, if, as Iamblichus says, the “most
perfect proportion,’ 12: 9 = 8: 6, was introduced by Pythagoras from
the apotome to harmony, as is stated by Eudemus, the Peripatetic.” Also, 2.17 p. 138:
“those who have written concerning these things declare that the Athenian Theaetetus
assumed two lines commensurable in square and proved that if he took between them
a line in ratio according to geometric proportion, then the line named the medial was
produced, but that if he took (the line) according to harmonic proportion, then the
apotome was produced.” On the concepts medial, binomial, and apotome, see Euclid
10; on the connection with the three means, Junge-Thomson 17. This fragment of
Eudemus is important because its content is independent of the Theaetetus of Plato and is
thus reliable evidence for the existence and importance of the mathematician Theaetetus.
It was cited by Sachs (135, 177), Frank (272; cf. above ch. V 2, n. 41), and von Fritz
(RE V A 1354f).
83 16. if one assigns the larger number to the Nete (above, ch. V I ἢ. 47): 6 is
Hypate, 12 is Nete, 8 (the harmonic mean) is Mese, and 9 (the arithmetic mean) is
Paramese.
84Jam. In Nic. 100.22 = DK 18.15: ἡ ποτὲ μὲν ὑπεναντία λεγομένη... ὑπὸ δὲ τῶν
περὶ ᾿Αρχύταν αὖθις καὶ “Inmagov ἁρμονικὴ μετακληθεῖσα, Also Archytas fr. 2, DK I
435.20f: τρίτα δ᾽ ὑπεναντία (péon), ἂν καλέοντι ἁρμονικάν; 1 436.8f: a δ᾽ ὑπεναντία,
ἂν καλοῦμεν ἁρμονικάν, In two other passages Hippasus and Archytas are named together,
in reference to the doctrine of means (Iam. In Nic. 113.16ff, 116.1ff). Tannery concluded
from this (MSc II 190, HScH 394) that Archytas must have cited Hippasus. Actually,
Iamblichus states in the two later passages, in contradiction to his own earlier account,
that Hippasus and Archytas had already discovered the fourth, fifth, and sixth means
(cf. n. 92); but it is possible that he himself misunderstood his source.—Philolaus, too,
knew the harmonic mean (A24).
85 Above, ch. V 1, nn. 33-34.
86 Hippocrates of Chios extends the idea of the geometric mean to that of the ‘‘two
mean proportionals” (not a: x = x:b, but a:x = x: y = y: Ὁ), in order to solve the
problem of the duplication of the cube; thus, even before his time, the geometrical
mean was known and used for “doubling the square,” i.e., for determining square
roots.
87 Music theory soon led, inevitably, to the series 6, 8, 9, 12. It is obvious that 9 is the
‘““mean’”’ between 6 and 12; but from a musical point of view it is just as clear that, as the
octave (6: 12) is divided by 9 into fifth and fourth, so it is divided by 8 into fourth and
fifth—drevavriws (cf. Philolaus fr. 6). Thus the expression ὑπεναντία μεσότης was
meaningful in music too.
88 Cf. C. Miiller, QSt 2 (1932) 281ff; Becker, MD 6s.
441
VE PYTHAGOREAN NUMBER THEORY
Babylon,’ we may be on the track of something genuine, namely
that the arithmetic and harmonic mean were used in Babylonian
calculating technique to find the square root.®® If so, a rule used in
Babylonian calculation has been transposed into Pythagorean number
speculation, just like the rule involved in the “Pythagorean theorem.”
The original name itself, ὑπεναντία μεσότης (“subcontrary”), is com-
prehensible from its use as a tool in calculation; one forms first the
“numerical mean,” then its appropriate “reversal.” Here too, then,
Pythagorean musical theory is an outgrowth of practical methods of
calculation. The further development of the doctrine of means only
begins with Eudoxus;* the accomplishment of the Pythagoreans
around and after Hippasus is rather in the application and interpreta-
tion of known methods than in the foundation of a theory of numbers.
An unquestionable piece of scientific Pythagorean arithmetic is
Archytas’ proof that a superparticular proportion cannot be divided
into equal parts by a mean proportional.® In contrast with the inductive
"Tam. In Nic. 118.23f: εὕρημα δ᾽ αὐτήν φασιν εἶναι Βαβυλωνίων καὶ διὰ Πυθαγόρου
πρώτου εἰς Ἕλληνας ἐλθεῖν. The proportion in question is, greater number: arithmetic
mican == harmonic mean: smaller number, or, greater number : harmonic mean — arith-
metic mean: smaller number (also, therefore, greater number Χ smaller number =
arithmetic mean x harmonic mean). The Babylonians did not know the concept of
proportion (Becker, Arch. βὶ Mus.-W. 1957, 156); but the ‘‘mean” may very well have
been employed in calculation.
Becker, MD ὅς.
%! Above, n. 9.
** He discovered the fourth, fifth and sixth means: Eudemus fr. 133 = Procl. In
Hucl. 67.$ff == D22 Lasserre; Iam. In Nic. 101.1ff (above, n. 84); Nicom. Ar. 2.28 and
Jam. fn Nic. 113.16ff only call the arithmetic, geometric, and harmonic means “ancient.”
Philip of Opus also wrote Meodryras (Suda s.v. φιλόσοφος). The seventh to tenth means
were discovered, according to Iam. In Nic. 116.4ff, by the Πυθαγορικοί Myonides and
Euphranor, not only later than Eudoxus but also later than Eratosthenes; not all Pythag-
orean arithmetic is pre-Platonic. It does not seem possible to discover any further detail
about Myonides and Euphranor (though a Euphranor wrote a Περὶ αὐλῶν, Ath. 4.184e;
cf, also Nicom. Ar. 2.28).—Thymaridas, whose ἐπάνθημα is mentioned at Jam. In Nic.
62.18f, has been placed “‘in the time of Plato or somewhat earlier” (Tannery, MSc
ΕΓ roaff, ΠῚ 1y2ff; HScH 396; also Becker, QSt 4, 165f; MD 43f), because the proposition
in question can be proved by a (complicated) pebble figure; but for the chronology this
docs not mean any more than a mere possibility. Heath, Math. I 94, lays down that the
terminology used about the ἐπάνθημα by lamblichus agrees with that of Diophantus
(probably 3rd century a.p.); and Diels (DK I 447 n.) emphasized that the definition of
the μονάς as περαίνουσα ποσότης, which is ascribed to the same Thymaridas by Iam. In
Nic, 11.2, cannot be dated earlier than Plato. Thymaridas is named as a pupil of Pythagoras
wt lim. VP 104, and edifying anecdotes about him are given at Iam. VP τάς (after the
“tripartitum’”’ 1.1... 8.9 ?) and 239; in the catalogue of Pythagoreans (p. 145.5) he is
listed as a Parian. How tradition and legend got connected with a specific mathematical
problem remains a mystery.
Ὁ) Archytas Arg -- Boeth. Mus. 3.11. See also ch. V.
442
2. Pythagorean Arithmetic
pebble games, a general proposition is here proved deductively, and
the numbers, represented by letters, are obviously thought of as
line segments, as in Euclid.** A number of arithmetical concepts, like
proportion, divisibility, the smallest numbers in a given ratio, relative
prime, and mean proportional, are confidently handled; the structure
follows conventional order: statement of what is “given,” statement
of theorem, proof;% and the method of proof is the reductio ad
absurdum. Above all, Archytas presupposes a whole scrics of arith-
metical propositions and expressly cites an auxiliary theorem.”
Tannery, who called attention to the proof of Archytas transmitted
by Boethius, concluded from this that Archytas must have had a kind
of Elements of Arithmetic,®” and van der Waerden undertook to recon-
struct, systematically, this number theory presupposed by Archytas,
coming to the conclusion that in all essentials the material of Euclid’s
seventh and eighth books must already have been in existence; book 7,
he thought, “existed in written form before 400 B.c.” and “had been
taken over by Euclid without significant alteration.”** Book 8 was
the work of Archytas himself.*? If this were correct, we should have
an imposing edifice of Pythagorean arithmetic of an entirely different
kind from what Speusippus, Aristotle, Theo, and Nicomachus lead us
to expect.
It is impossible to discuss in detail here the analysis of Euclid and
the problem of evaluating his work as a mathematician ;1"° but we may
94 Unlike Euclid, the author designates with letters not the ends of line segments
(“line AB”), but the segments themselves; DE means the number D plus the number E.
This is not, however, an ancient feature (as maintained by Tannery, and van der Waerden,
MtAnn 1947-1949, 134), but the method followed by Boethius (also Mus. 4.2, in the
translation of Sect. can. 3). Every exposition in the history of mathematics “ modernizes.
95 To be sure, the structure can be the work of the intermediate source. . . .
96 Archytas A1g: ‘“. . . qui enim sunt minimi in eadem proportione, quibuslibet allis
numeris, hi primi ad se invicem sunt,”’ corresponding to Eucl. 7.22.
97 Tannery, MSc ΠῚ 249; cf. Heath, Math. I 90; Eucl. II 295; Becker, MD 44f.
% MtAnn 1947-1949, 146.
99 Ibid. 149; cf. SA 153ff, RE XXIV 284-285. ο΄
100 Van der Waerden, SA 197: ‘‘Euclid is by no means a great mathematician,
though he is “τῆς greatest schoolmaster known in the history of mathematics (196).
This judgment calls for contradiction; the Elements does constitute a scientific achieve-
ment, rather than merely a schoolbook. And it is based in part on a mistranslation. In
the scholium to Euclid, p. 654.8 Heiberg, one reads, after the account of the discovery of
the regular solids by the Pythagoreans and Theaetetus, Εὐκλείδου δὲ ἐπιγράφεται καὶ
τοῦτο τὸ βιβλίον διὰ τὸ στοιχειώδη τάξιν ἐπιτεθεικέναι, “This book too [sc. 13] bears
Euclid’s name, because he gave it the order of στοιχεῖα (see Burkert, Philologus 1959,
1806). Cf. Schol. Eucl. p. 73.4 (after Proclus 69.4ff), συνήγαγεν εἰς στοιχείωσιν, τάξιν
αὐτοῖς καὶ ἀποδείξεις ἀκριβεστέρας ἐπιθείς... Van der Waerden, however, translates
(SA 173), “This book also carries Euclid’s name, because he embodied it in the Elements,
as though he had taken it over from an earlier author without alteration or addition.
443
Fy ἅι 5. ὁ αὶ ἃ δια ee EET EV WEE SEEN δ 2 SENN OE
refer once more to the large number of mathematicians who were
active in the brief period between Plato and Euclid! Several of these
were concerned, precisely, with writing or rewriting στοιχεῖα. 02 The
miracle of Pythagorean arithmetic would indeed be amazing, if in
spite of this, Euclid had taken over ‘without significant alteration”’
expositions that were over a hundred years old.
Here we are confronted again with a basic problem of mathematical
history: to be sure, it can be shown with perfect exactitude what
propositions Archytas must have been presupposing as proven—but.
only on the presupposition, on our part, that Archytas built up a
complete theory of number, in accordance with the demands of
Euclidean precision. Even the suggestion that Archytas was concerned
to prove a proposition in number theory is a dubious formulation.
Bocthius cites the proof in the context of music theory; and among
the works of Euclid, it is found in the Sectio canonis, not in the number
theory of the Elements. There is no question that for Archytas the
musical application of the theorem—the indivisibility of the intervals
of the octave, the fifth, the fourth, and the whole tone—was of
primary importance. To what extent he was able to refer to a pre-
viously existing arithmetic, and to what extent he understood and
fulfilled the demands of a complete, deductive system, remains to be
established.
Scholars have laid emphasis on the fact that Archytas’ proof was
identical with the one found in the Sectio canonis (3) Boethius,
however, characterized Archytas’ proof as “nimium fluxa” and
replaced it with another, consisting of a literal translation of the
Sectio canonis+®* Thus the tradition which Boethius was following
cmphasized not so much the agreement as the difference between
Archytas’ proof and that of the Sectio canonis and found Archytas’
inadequate. In order to understand Boethius’ criticism, we must
compare the two proofs in detail.
The first step they have in common: for the given superparticular
proportion a:b is substituted that of the two smallest numbers in
101 Above, ch. VI 1, n. 125.
102 Eudemus at fr. 133 = Procl. In Eucl. 66.20 names Leon, at 67.14f Theudius, and at
67.20ff Hermotimus.
13'Van der Waerden, Hermes 1943, 169, and SA 153, concludes from this that like
Euclid 8, the Sectio canonis mainly derives from Archytas (i.e., without the last 2 sentences;
above, ch. V 1, π. 22), while Frank sees in the musical theory of Archytas the sharpest
kind of contrast to the “Platonically” influenced Sectio canonis (above, ch. V I).
104. Boeth. Mus. 4.2 p. 303.19ff. The whole passage, Sect. cat. 1-9, is translated by
Boethius at Mus. 4.2.
444
x ΞΞ αἱ ὁ ae = Se ΨΥ Ὁ ὙΠ τὸ εν
the same ratio; min παν θν ‘The proof continues, inthe ΔΘ
canonis, as follows: (1) mand n are by definition, ax the smallest
numbers in the given ratio, prime to each other. (2) According, to
the definition of the superparticular proportion, the difference between
m and ἢ is a common divisor of m and ἢ. (3) Thus m -τ ἢ “61.
(4) Accordingly there is no mean proportional between m and ἢ,
(s) and by extension there is none between any numbers that stan
in the same ratio, a: 0.196 Archytas, for his part, first establishes step
(2), mistakenly introducing, in the statement of the reason, the concept
“ minimi.’1°? Then he sets up proposition (3), as an assertion and proves
it in detail, by reductio ad absurdum of the opposite. This is a basi-
cally superfluous repetition,!°* made necessary by his view that τ is not
a number.° The only crucial statement is (1), which is cited in full.
After another repetition,”° steps (4) and (5) follow without detailed
rationale; the necessary, and not quite evident, auxiliary theorem is not
cited. 7
The repetitions here could be regarded as no more than a stylistic
defect, but the fact that necessary and nonnecessary presuppositions
are not distinguished strictly is more disturbing ; and for this reason we
can hardly suppose that it is only some copyist’s fault that the necessary
auxiliary theorem is not cited. A particularly arresting formulation is
found in the citation, in Euclid, of another lemma: “qui enim sunt
minimi in eadem proportione quibuslibet aliis numeris, hi primi ad se
invicem sunt, et solam differentiam retinent unitatem. 1? Tannery
excised the last five words as “absurd,’’"* and the more recent para-
phrases leave them out;”* but the criticism of Boethius, that is, that
105 This depends on Eucl. 7.22.
106 Citing Eucl. 8.8. 7
107 DK 1 429.35: “‘quoniam igitur sunt minimi in eadem proportione C, DE et sunt
superparticulares . . .” Only their superparticularity is necessary for the conclusion.
108 DK I 429.36: “ΠῈ numerus C numerum parte una sua ¢ciusque transcendit; ea
haec D”; then 429.38: “‘si enim est numerus D et pars est eius qui est DE, metietur
numerus DE numerum, quocirca et E numerum metietur; quo fit, ut C quoque metiatur,
igi tietur ...”
utrumque igitur, C et DE numeros me ΠῚ
109 DK I 429.37f: “ἀΐςο quoniam [ὅτι] D [i.e., m — n] non erit numerus sed unitas.
The proof of irrationality in Eucl. 10 App. 27 proceeds in a quite similar manner.
110 DK I 430.5: “igitur DE numerus C numerum unitate transcendit. sc
111 J... Eucl. 8.8; cf. n. 106. In the list of propositions presupposed by Archytas whic
van der Waerden sets up (SA 111), the auxiliary proposition Eucl. 8.8 is one of the most
important links.
112 DK I 430.3; cf. Eucl. 7.22.
18 MSc Il 248. (He thinks this is an interpolation by Boethius; but shall we suppose
he made interpolations only to argue against them?)
114 F.¢, Heath, Math. 1 215; van der Waerden, SA III.
445
VIL PYTHAGORBAN NUMBER THEORY
of his Greek source-—-confused as it is takes this very sentence as its
point of departure.” The ditlerence of the “smallest numbers in a given
ratio” is not of course 1 in every case, but only in the superparticular
ratio. Did Archytas perhaps, as he previously introduced the idea of
“minimi” when dealing with the superparticular ratios (above, ἢ.
107), think of superparticulars when dealing with “minimi” and so
formulate a conclusion which was only applicable to the former?"
In any case, athctesis is no suitable way to get rid of the problem. This
is to ercate the illusion of a perfect mathematical clarity and obstruct
our view of a mathematics still feeling its way along an uncertain
path.
Archytas’ achievement is not to be underestimated. A start was made
on the way to a general theory of numbers, such as we find in Euclid;
but it is certain that the perfection of Euclidean form had not yet been
reached, and that the similar proof in the Sectio canonis was regarded
as a basic improvement. Archytas’ number theory grew out of his
music theory, and a complete, systematic structure is not to be expected.
Was Archytas the first, using the methods of proof suggested by a
highly developed geometry, to create a deductive theory of number of
which Philolaus still had no suspicion?” The name “number theory,”
ἀριθμῶν θεωρία,118 appears to come from Xenocrates, following Plato
in his demand for a “ pure,’ logical and deductive treatment of numbers,
above and beyond the realm of sense Impression—éa τῆς τῶν ἀριθμῶν
φύσεως 11) Thus the number theory stemming from Archytas—like,
1 Bocth. Mus. 3.11 Pp. 286.7ff (no longer included in DK): “Et secundum Archytae
quidem rationem idcirco in superparticulari nullus medius terminus cadit, qui aequaliter
dividat proportionem, quoniam minimi in eadem Proportione sola differunt unitate,
quasi vero non etiam in multiplici proportione minimi eandem unitatis differentiam
sortiantur, cum plures videamus esse miultiplices praeter eos, qui in radicibus [πυθμένες =
minimi, Archytas A17] collocati sunt, inter quos medius terminus scindens aequaliter
candem proportionem possit aptari. Sed haec, qui arithmeticos numeros diligenter
inspexerit, facilius intellegit. Addendum vero est, id ita evenire, ut Archytas putat, in
sola superparticulari proportione; non autem universaliter est dicendum,” Perhaps
Boethius did not himself understand his ‘“reprehensio”’ (285.8). The concluding sentence
is correct; what Archytas (apparently at least) expressed in general terms, that the ‘“‘differ-
entia”’ of the “minimi” is one, is valid only for the superparticular proportion.
"It would also be possible that an intermediate source transposing Archytas’ proof
into the later form created the confusion; or that archaic terminology was misunderstood,
and Archytas meant, with a phrase like ὑπέχειν τὴν μονάδος διαφοράν, the “common
divisor 1" (the common divisor is found by alternate subtraction).
"7 Above, ch. V 2. But the title ᾿Αριθμοί is also found in Democritus (B1I 0).
"The programmatic title of a work of Xenocrates (1).L. 4.13).
PL Rep. 5250; on the subject in general, aside from Rep. ς22ς et seq., Phib. 56d,
Tht. tosd, pin. ggoc. When Plato, at Rep. §25¢, cites of περὶ ταῦτα δεινοί, the “Ργοΐος--
sionals,” for their axiom that the monad is indivisible and fractions are not allowed in
the calculation of proportions, he may well mean the same Pythagorcans he speaks of at
446
.. Pythagorean Geometry and Mathematical Secrets
ina sense, Pythagorean philosophy in’ general-—achieved its. final
form in Platonism., Before Archytas there were number games
: . wre tt τ . :
accompanied by the “interpretation” of, and “reverence” for, number,
3. PYTHAGOREAN GEOMETRY AND MATHEMATICAL SECRETS
A passage of Iamblichus, to whose Aristotelian provenance we have
already drawn attention, reads as follows:1
of δὲ Πυθαγόρειοι διατρίψαντες ἐν τοῖς μαθήμασι καὶ τό τε ἀκριβὲς
τῶν λόγων ἀγαπήσαντες, ὅτι μόνα εἶχεν ἀποδείξεις ὧν μετεχειρίζοντο
ἄνθρωποι, καὶ ὁμολογούμενα ὁρῶντες [ἔνισον] τὰ περὶ τὴν ἁρμονίαν,
ὅτι δι᾿ ἀριθμῶν, [καὶ τὰ περὶ τὴν ὄψιν μαθήματα διὰ «δια »γραμμάτων],
ὅλως αἴτια τῶν ὄντων ταῦτα φήθησαν εἶναι καὶ τὰς τούτων ἀρχάς" Sore
τῷ βουλομένῳ θεωρεῖν τὰ ὄντα πῶς ἔχει, εἰς ταῦτα βλεπτέον εἶναι, τοὺς
ἀριθμοὺς [καὶ τὰ γεωμετρούμενα εἴδη τῶν ὄντων] καὶ λόγους, διὰ τὸ
δηλοῦσθαι πάντα διὰ τούτων.
The Pythagoreans, having devoted themselves to mathematics, and
admiring the accuracy of its reasonings, because it alone among
human activities knows of proofs, and seeing [equally] the facts
about harmony, that they happen on account of numbers, generally
admitted [and (seeing) the mathematics of optics depending on
(dia)grams], they deemed these (facts of mathematics) and their
ni it Θ Ῥἅ'
Rep. 530, i.e. the circle of Archytas. A μάταιος πραγματεία in relation to λογισμοί is
also mentioned by the Xenophontic Socrates (Mem. 4.7.8).—The assertion so often
repeated, that Plato differentiated λογιστική, as the practical art of calculation, from
ἀριθμητική, as the theory of number, and that the latter was an achievement of the
Pythagoreans (Tannery, HScH 381; Heath, Math. I 13f; Dodds, Gorg. 199; K. Vogel,
“ Beitrage zur griechischen Logistik,” SbMii, math.-ph. KI., 1936, 361f; Becker, MD 45),
is an error long ago corrected (J. Klein, ‘‘Die griechische Logistik und die Entstehung der
Algebra,” QSt 3 [1936] 23ff; A. Wedberg, Plato’s Philosophy of Mathematics [Stockholm,
1955] 22f). Plato demands mathematics as pure theory for λογιστική as well as for
ἀριθμητική, and there is a practical “arithmetic” as well as a theoretical art of calcula-
tion,” or “logistics.” (Phlb. 56e). The difference between arithmetic and λογιστική
corresponds to that in everyday language between “counting” and “calculating.
(Charm. 166a, Gorg. 451b; cf. Rep. 522e, Epin. 978e; the two together, Rep. 522c~e,
5258, Phdr. 274c, Hp. mi. 367a; λογιστικοί alone, Euthyd. 29ος, Rep. §25b, Pol. 259%,
Tht. 145a; cf. also Rep. $10c: of περὶ... λογισμοὺς... ὑποθέμενοι τό τε περιττὸν καὶ
τὸ ἄρτιον... with Tht. 198a-c: ἀριθμητική as the ἐπιστήμη ἀρτίου καὶ περιττοῦ). The
differentiation of theoretical arithmetic and practical “logistics” was set up by Geminus
(by contrast to the “Pythagorean” quadrivium: Procl. In Eucl. 38.1ff; Geminus-
Anatolius ap. Hero Def. 135.5 p. 98 Heiberg). Olympiodorus foisted it onto Plato
(Olympiod. In Gorg. 450} p. 31.4ff Norvin; cf. Schol. Charm. 165e).
'Tam. Comm, math. sc. Ὁ. '78.8-18; cf. above, ch. 1 2, n. 112.
447
Yi. FPYTHAGORSAN NUMBER THEORY
principles to be, generally, causative of existing things, so that
whoever wishes to comprehend the true nature of existing things
should turn his attention to these, that is to numbers [and the
geometrical entities among existing things] and proportions, because
it is by them that everything is made clear.
In the two places where geometry is mentioned the course of the
argument is deranged. Ὁ begin with the second, λόγους (““ propor-~
tions’’) is without an article, and, as regards the sense, is intrusive
after the clearly established dichotomy of arithmetic and geometry;
and amidst the Aristotelian formulations the phrase τὰ γεωμετρούμενα
εἴδη τῶν ὄντων sounds like late Platonism.? To delete the offending words
gives τοὺς ἀριθμοὺς καὶ λόγους, which belong together and suit the
context. In the earlier passage, too, there are several difficulties. The
word ἔνισον is peculiar and unexampled,? and ὅτι destroys the conti-
nuity.* τὰ περὶ τὴν ὄψιν μαθήματα is not only unclear,® but factually
wrong; it is not the fact that mathematics is mathematical or geometri-
cal that moved the Pythagoreans so deeply, but that an everyday
concern like music, impinging on us directly through the senses, turns
out to conform to mathematical rules. Thus τὰ περὶ THY ὄψιν μαθήματα
is not truly analogous to τὰ περὶ τὴν ἁρμονίαν. Again, to delete the
reference to geometry not only makes ἔνισον superfluous, but gives a
sinoother and more meaningful sentence. Aristotle, in the Metaphysics
(985b31), names the connection of music and number as one of the roots
of Pythagorean speculation, but never brings this into relation with
geometry.® Aristoxenus, too, in the one relevant passage remaining
to us (fr. 23), speaks only of “concern with numbers,” which Pythagoras
especially “honored” and advanced. It seems likely, then, that Iam-
blichus, in a passage drawn from Aristotle, added the references to
geometry, so as to emphasize the many-sidedness of Pythagorean
μαθήματα, whereas in the original testimony geometry plays no role,
in contrast to that of arithmetic.
"ra γεωμετρούμενα is a school term from late antiquity. See Plut. Quaest. Plat. too1-
"002, Proch In Bucl. 211.4, Hero Deff. τ p. 14.19, 136.51 Ρ- 150.17, Geom. 2 p. 176.1
ΕΠ να.
* Vitelli suggests ἐπ᾿ ἴσον. Perhaps ἐξ ἴσου
* Pesta deletes ὅτι.
* Aristotle classifies ὀπτικά with gcometry (as apzovxd with arithmetic: An. post.
7516, Met. 997b20, Met. 1078a14, An. post. 78b37, Phys. 194a8ff). He speaks of “ geo-
inetrical optics’; but he never mentions Pythagoreans in this connection.
"It has been shown above (ch. I 2, n. 75) that the doctrine of πέρατα was
not Pythagorean,
448
1, Pythagorean Geometry and Mathematieal Secrets
The most important achievement that we know of that was made
by a Pythagorean in the field of geometry is the famous solution, by
Archytas, of the problem of doubling the cube.’ Successful accomplish-
ment of this task, talked of for at least a gencration but never yet
performed, stimulated further action. The improved solutions, from
Eudoxus and Menaechmus to Eratosthenes, do not diminish but
confirm the credit of the πρῶτος edperjs. But the mathematician who
was Archytas’ immediate predecessor in this regard was Hippocrates
of Chios. It was he who had reduced the problem of doubling the
cube—or finding ~/72—to that of the “two mean proportionals,”
and thereby given it a planimetric sense which first made the solution
possible.* Thus Archytas, in his decisive geometrical achievement, is
not the representative of a special, Pythagorean tradition, but of the
main tradition of Greek geometry in general, represented in his time
by Hippocrates of Chios; and, if he was successful, it was probably
mathematica quadam facultate, non pythagorea.
The real problem of Pythagorean geometry, therefore, in the
history of thought and of mathematics is this: are we to assign the
well-attested geometrical accomplishments of Pythagoreans to the
time before or after Hippocrates of Chios? Only in the former case
could we speak of the Pythagoreans as founders of Greek geometry; in
the latter, it is only a matter of individual contributions to the develop-
ment and perfection of something already there, quite respectable
achievements but on an equal basis with others, and themselves
dependent on older and more basic work. Only Heidel put the problem
in this fashion;® “Pythagorean geometry” is usually treated, as
though this were perfectly natural, before Hippocrates. It is assumed
that he had before him at least the content of the first four books of
Euclid, and that these are of Pythagorean origin.!° This is in spite of
the fact that the tradition names Hippocrates as the first author of
στοιχεῖα.
Citing Eudemus, Proclus refers two passages in the first book of
Euclid to the Pythagoreans—though not to Pythagoras. These are the
7 Archytas A14 = Eudemus fr. 141 = Eutocius In Archim. II? 84 Heiberg. On the
problem of duplication of the cube, see Heath, Math. I 244ff; Becker, MD 75ff. To be
sure, Archytas’ construction cannot be executed with ruler and compass.
8 Hippocrates, DK 42.4 = ps.-Eratosth. Epist. ad. Ptol. ap. Eutocius In Archim. TI?
88.17 Heiberg; also Procl. In Eucl. 213.7ff (not in DK).
® Heidel, AJP 1940.
10 Tannery, Géom. 106ff (cf. above, ch. VI 1, n. 44); Rey 240 (despite his skepticism in
principle, 230ff); van der Waerden, SA 135; with reserve, Heath, Math. 1 τόσες
449
yr me elon ewe Vere ae YY ΒῪ ere eee ἃ δάόκΜΝΤκ EN ἃ
proof for the sum of the angles of a triangle! and the method of
application of areas." The same Kudemus may be the source of a few
further reports about discoveries of the Pythagorcans—not of Pythag-
oras—in Proclus and the scholia to Euclid3 The whole fourth book
of Euclid is Pythagorean," as is the proposition that 6 equilateral
triangles, or 4 squares, or 3 regular hexagons, will fill the plane about
a given point; and the three regular solids tetrahedron, cube, and
dodecahedron'® (the two others, octahedron and icosahedron, are
ascribed to Theactetus). A common origin for the tradition about all
these propositions is suggested by their close interdependence. The
content of the fourth book is the relationship of the regular polygon
and the circle; the exhaustion of the plane by regular polygons pre-
supposes the proposition about the sum of the angles of a triangle, and
leads naturally to the construction of regular polyhedra; and the solution
of quadratic equations by the application of areas is a presupposition
of the construction of the regular pentagon as well as of further
problems in the construction of polyhedra.
It is very likely that these data are incomplete. Proclus’ detailed
commentary on Euclid is extant only for the first book, as is true of
Pappus’ commentary on book τὸ (in Arabic translation). The scholia,
which of course draw from the commentaries, have a habit of omitting
just those details which are useful for the history of mathematics; so
that it may have happened that Eudemus’ reports of Pythagorean
discoveries in the other books of Euclid have been lost. On the other
hand, the Pythagoreans, as predecessors of the “divine” Plato, were
extremely important for Proclus, who regarded the word μαθηματική
itself as Pythagorean in origin.!? Therefore he would surely not have
ἰοῦ slip any opportunity to mention Pythagorean achievements. This
authorizes us to surmise that Eudemus, and the intermediary sources
" Procl. In Eucl. 379.2 = Eudemus fr. 136 = DK 58Ba1 (Eucl. 1.32).
Procl. In Eucl. 419.15 = Eudemus fr. 137 = DK 58B20 (Eucl. 1.44); cf. Iam. Comm.
math. sc. p. 75.21, Plutarch, above, ch. VI 2, ἢ. 7.
18 Tt can be shown that the scholia to the first book are mostly from Proclus (cf. e.g.
above, ch. VI 2, n. 6), and those to the tenth book from Pappus (cf. e.g. below, ἢ. 57).
Both cite Eudemus. |
“’ Schol. Eucl. 273.3: εὕρημα δὲ τοῦτο τὸ βιβλίον τῶν Πυθαγορείων, 273.13: τὰ ὅλα δὲ
θεωρήματα τοῦ προκειμένου βιβλίου ιζ΄ (read ts’) ὄντα Πυθαγορείων εὑρήματα. This
testimonium is missing in DK, and not mentioned by Heath (Math. I), van der Waerden
(SA), or Becker.
16 Procl. In Eucl. 304.11 (cf. Arist. Cael. 306b4, Eucl. 1.1 5, porism).
; © Schol. Eucl. 6$4.3: τρία δὲ τῶν προειρημένων ε΄ σχημάτων τῶν Πυθαγορείων ἐστίν,
ὅ τε κύβος καὶ ἡ πυραμὶς καὶ τὸ δωδεκάεδρον, Θεαιτήτου δὲ τό τε ὀκτάεδρον καὶ τὸ
εἰκοσάεδρον. Cf. ch. I 3, n. 116; on χροιά, “plane surface,” see ch. I 3, n. 96.
” Procl. In Eucl. 45.5ff; cf. above, ch. VI 1, n. 119.
450
@* . ἢ grea eee’ ἌἯεΝ τς δ ν ἡ ψέεννν ὩΨΙΕΌΥΨΕΕΨΥΨΥΥΥ se eee eee eee
for history of mathemates dependent on him! did not lave anything
more to say about Pythagorcanism in the first book of Bachad. Pus in
turn gives us certain clues to the evaluation of Pythagorean contrabu-
tions in the Euclidean corpus: there are two Pythagorean theorems in
the first book, and other propositions, proofs, and constructions are
ascribed to Thales or Ocnopides. The fourth book is Pythagorean, as
the fifth is fron. Eudoxus.1® What we are concerned with, then, is
certain individual, restricted portions, to be ranked along with what
other mathematicians “discovered”; there is no reason to posit a
unique contribution by the Pythagoreans or a substratum of Pythag-
orean Elements. More than once, before Euclid, Στοιχεῖα had been
written, and each time their structure and methods of proof had been
thought through again.?°
The chronology of these Pythagoreans is not guaranteed by any
external testimony;?! whether we go up to the early fifth century or
down to the fourth, everything depends on combinations. Since
Theaetetus, who was still very young in 399 B.c. and fell in battle in
369 B.c.,22 brought to completion the treatment of the regular poly-
hedra, the “Pythagoreans’’ in question must be placed earlier. Eudemus
names Theaetetus, Archytas, and Eudoxus, mentioning their individual
accomplishments,” so that the anonymous Pythagoreans probably
belong to the time before Archytas. The decisive question of their
18 Proclus surely did not use Eudemus directly. Tannery (Géom. 18ff) concluded that
Geminus was the intermediary.
19 Schol. Eucl. 280.7ff: τὸ δὲ βιβλίον Εὐδόξον τινὲς εὕρεσιν εἶναι λέγουσι...
20 The basic stock probably goes back to the Στοιχεῖα of Hippocrates of Chios, which
were the first of their kind (Eudemus fr. 133 = Procl. In Eucl. 66.7: πρῶτος yap ὁ
“πποκράτης τῶν μνημονευομένων καὶ στοιχεῖα συνέγραψεν). Von Fritz shows (ABG
1940, 72) that the first books of Euclid underwent a thorough revision after the time of
Aristotle, perhaps at the hands of Euclid himself.
21 Procl. In Eucl. 419.15 = Eudemus fr. 137: ἔστι μὲν ἀρχαῖα, φασὶν of περὶ τὸν Εὔδημον,
καὶ τῆς τῶν Πυθαγορείων μούσης εὑρήματα ταῦτα. This sounds as though Eudemus had
been speaking of “‘ancient” Pythagoreans; but Proclus’ next words, ἀπὸ δὲ τούτων καὶ
οἱ νεώτεροι... refer to Apollonius of Perga. Thus the word ἀρχαῖα comes from Proclus
(or Geminus), for whom the Pythagoreans are simply, as a group, “ancient” (cf. 419.21f,
420.22f).—When von Fritz writes, on Eudemus’ fragment 136 (AnnMath 1945, 258),
‘“‘Eudemus . . . attributes to the early Pythagoreans the proof...” the word “early”
does not correspond to anything in the Greek text.—According to Aristoxenus the
“last Pythagoreans” were living about 366/365 B.c. (above, ch. II 5, n. 43).
22 Cf, Sachs 88ff; von Fritz, ΚΕΝ A 13$2f.
23 Theaetetus, above, ch. VI 2, n. 82; Archytas: Eudemus frr. 60, 65, 141; Eudoxus:
Eudemus frr. 133 and 148. The indefinite expression ‘‘Pythagoreans” could have the
same explanation in Eudemus as for Aristotle (above, ch. I 4): where he cautiously uses
this expression, the persons in question themselves referred to ‘“‘ Pythagoras.” But he
could also mean Archytas; cf. above, ch. I 2, n. 106, on Eudemus fr. 60. It is also con-
ceivable that Theodorus of Cyrene is at least included in the expression (above, ch. VI 1,
451
ΓΑΙ 8 8 δ Ἀ “ΔΕΥ ΕΥ̓ ΒΕΗ͂Α ΓΡΙΠΌΚΥ
relation to Hippocrates of Chios depends on inner criteria, on the
relation of concepts, propositions, constructions which seem to be
presupposed by the Pythagoreany or by Hippocrates. In a closed
system like geometry, so many combinations of the same propositions
are possible, and so many proofs for the same conclusions, that it is
never possible to establish chronological priority with mathematical
certainty,
The application of areas was known to Plato,24 but Hippocrates of
Chios, for a problem soluble by this method, used the method of
“inclination” or “verging” (νεῦσις); it looks as though the application
of areas was at least not fully developed in Hippocrates.” A special
case of the application of arcas is the construction of the golden section.
The star-pentagon or “pentagram” and the dedecahedron play a role
in Pythagoreanism, but both figures had been known, in a purely
empirical way, for centuries,2® without any mathematical construction.
Euclid gives the construction of the golden section, and therefore of the
regular pentagon, in book 4, which is ascribed “as a whole”’ to the
Pythagorcans; and it depends on constructions in book 2. Hippocrates
eee
n. 12). In spite of this, Frank’s formulation is too sweeping : “What is ordinarily called
Pythagorean mathematics is in essentials the work of Archytas, Theaetetus, and Eudoxus”
(232). Theactetus is never called a Pythagorean, and Eudoxus is treated in the tradition
as an original thinker.
# On the idea and the method of the application of areas, see Heath, Math. I 1 soft;
Becker, MD 6off; van der Waerden, SA 118ff. Euclid 2 has a generalized and therefore
more complicated form of the application of areas.—rapareiveww, Pl. Rep. 5274; a specific
problem, Pl. Meno 86e (cf. Heath, Math. I 298ff; Becker, Gnomon 28 [1956] 225). διορισμός
to which Plato alludes here, is a method developed by Leon, one of his contemporaries
(Eudenius fr. 133 = Procl. In Eucl. 66.22). Just as Plato learned from Eudoxus in the field
of astronomy (above, ch. IV 2), so in mathematics he learned from his contemporaries,
hot from “ancient” Pythagoreans.
4 Heller, AbhBln 1958, 9. comes to the conclusion that Hippocrates still did not know
the method of application of areas. See also Becker, MD sof.
" Above, ch. II 4, n. 81; below, n. 65. That the magical efficacy of the pentagram does
not depend on mathematically exact construction is shown by the fact that they are often
drawn quite irregular. See, e.g., the Gallic coins in W. Deonna, Bull. de l’Ass. Pro Aventico
16 (1054) 47; he also mentions (p. 24 n. 4) an empirical method of casting metallic dodeca-
hedra, with no recourse to Euclidean geometry.—Heath conjectures that the “golden
section’ was a discovery of the Pythagoreans (Math. I 160f, Eucl. 2.907); cf. Becker,
MD) 63. Heller, AbhBin 1958, οἵ, tries to reconstruct an earlier νεῦσις construction of
the regular pentagon. In Eudemus fr. 1 33 = Procl. In Eucl. 67.6 we read, (Εὐδοξος) τὰ
περὶ τὴν τομὴν ἀρχὴν λαβόντα παρὰ Πλάτωνος εἰς πλῆθος προήγαγεν καὶ ταῖς ἀναλύσεσιν
ἐπ᾿ αὐτῶν χρησάμενος. Sachs (07, 1286), Sarton (Hist. 442f), and Heller (AbhBln 1958, 12£ )
interpret this as referring to the golden section, which would imply that the report of
the scholium on Eucl. 4 (above, n. 14) was false. But the formulation in Proclus is so
general that this conclusion does not hold up. The topic may be the new, systematic
development based on Eudoxus’ new theory of proportion (as Heller thinks), or it could
be the systematic development of Eucl. 2. In the first 11 of the 14 propositions of the
second book the problem is the division (τέμνειν) of a linc!
452
.. Pythagorean Geometry and Mathemathal Secrets
surcly knew essential parts of the second book,® but not tecessarily
the content of the fourth. Plato seems to know the golden section!
The proposition about the angles of a triangle seems to be simple
and fundamental; Aristotle knows it in the “Euclidean” form, differing
in a small detail from the Pythagorean;?® still, in what follows in
Euclid nothing depends on it except the proposition about the occupa-
tion of a plane about a point by regular polygons (above, n. 15). It
obviously became important in the time of Aristotle for the fundamen-
tal questions of geometry, which lead to the formulation of the
Euclidean parallel postulate.8° The relation of Eudemus’ “ Pythag-
oreans’’ to Hippocrates remains ambiguous.
There is no such ambiguity in the comparison with the achievements
which Eudemus ascribes to Oenopides. These constructions are so
elementary that Tannery concluded Pythagorean geometry must
really have been a strictly guarded secret of the school, and therefore
inaccessible to Oenopides.*! It is stated that Ocenopides stole from
Pythagoras the determination of the ecliptic, which shows that he must
have measured its angle. The usual approximative value is the arc of a
regular fifteen-angled figure;3? and it is with the construction of such
a polygon that the fourth book of Euclid, whose origin is Pythagorean,
closes. This presupposes Oenopides’ result; Pythagorean achievements
in geometry are later than Oenopides, and also hardly earlier than
2? Namely Eucl. 2.12-13 (the extension of the ‘Pythagorean theorem” to scalene
triangles) and 2.14 (the transformation of rectangle into square, construction of mean
proportionals, and the geometrical construction of square roots)—the culminating
propositions of the second book.
28 The simplest interpretation of the mathematical allusion at Hp. ma. 303b relates it to
the golden section (Heath, Math. I 304; more complicated: E. Strycker, Mélanges Boisacq
{Paris, 1937] 317-326).
59 Heath, Aristotle 23ff, on Arist. Met. 1051424, ἀνάγειν. The necessary auxiliary line
is drawn from the base of the triangle upward in Eucl. 1.32, in Eudemus fr. 136 through
the apex of the triangle and parallel to the base.—Geminus (Eutocius on Apollonius,
II 170.4 Heiberg) states that the proposition about the sum of the angles of a triangle
was first proved separately for equilateral, isosceles, and scalene triangles, πρότερον ἐν
τῷ ἰσοπλεύρῳ παὶ πάλιν ἐν τῷ ἰσοσκελεῖ καὶ ὕστερον ἐν τῷ σκαληνῷ. This may be spun
out of a passage of Aristotle, which mentions this merely as a logical possibility: οὐδ᾽
ἄν τις δείξῃ καθ᾽ ἕκαστον τὸ τρίγωνον ἀποδείξει ἢ μιᾷ ἢ ἑτέρᾳ ὅτι δύο ὀρθὰς ἔχει ἕκαστον,
τὸ ἰσόπλευρον χωρὶς καὶ τὸ σκαληνὲς καὶ τὸ ἰσοσκελές, οὔπω οἷδε τὸ τρίγωνον ὅτι δύο
ὀρθαῖς... (An. post. 74a25ff). See Heath, Math. I 136, Eucl. I 310f, following Heiberg.
But Becker has shown that the individual proofs can be grasped easily and clearly from
considerations of symmetry (Grdl. 27, MD 39).
0 See Heath, Math. I 339, 358, 375, Ἐμοὶ. I 191; I. Téth, ‘Das Parallelenproblem im
Corpus Aristotelicum,” Arch. for Hist. of Exact Sciences 3 (1966) 249-422. Archytas
addressed himself to the problem of the infinity of space (Eudemus fr. 65).
“1 Tannery, Géom. 86; DK 41.12-14 (Eudemus fr. 138); above, ch. VI 1, nn. 128-130.
32 T.e. 24° (correct value, 23° 27’ 17).
44}
yr BF ase ROMANS ANGEEY ἐΥΒΗΙΚς I FIBUK Y
Hippocrates of Chios. "They belong in the period, Approximately,
between 430 and 400. mess it remains obscure what individuals are
lurking beneath the general appellation of “Pythagorcan” (above,
Nn. 23),
The most important “discovery” was that of the application of
areas. Scholars now agree that the point of these exercises is primarily
algebraic; they provide an cquivalent for quadratic equations. In
Babylonian mathematics they had been solved algebraically, and the
individual examples of the application of areas correspond exactly to
the methods developed there. Thus the “geometrical algebra” of the
Greeks is revealed as the transposition of Babylonian techniques of
calculation into geometrical form.* The occasion for the development
of this seemingly complicated “ geometrical algebra” was the dis-
covery of the irrational, which made impossible, according to Greek
views, the solution of quadratic problems by use of number. In fact, the
dicovery of the irrational is also ascribed to Pythagoreans, or even to
Pythagoras himself; but the actual situation is extremely hard to grasp
because of the profusion of ancient legend and allegory, and the modern
conjectures they have inspired. The ancients speak of this situation in
terms of “secrecy” and “treason,” the moderns of the “ Grundlagen-
krisis der griechischen Mathematik.”
Pythagorean lore was at least in part secret, as is attested by Aristotle
and Aristoxenus, and as is natural in an archaic social order.35 On the
other hand, Pythagorean “secrecy” was undoubtedly misused in
later times, as a carte blanche to permit the publication of forgeries as
newly discovered books, and brand the discoveries of later thinkers as
plagiarism of Pythagoras. Names that occur in such context are
Empcedocles,?? Philolaus,38 Oenopides,?® a certain musical theorist
33 This was Heidel’s conclusion, AJP 1940, 18.
34 Fundamental is Neugebauer, QSt 3, 245ff: cf. ExSc 147ff; 5. Gandz, Osiris 3 (1938)
460-470; van der Waerden, MtAnn 1947-49, 131f; SA 124f; RE XXIV 286-288; above,
n. 24. The expression “geometrical algebra” has been current since Zeuthen. Van der
Wacrden calls the Pythagoreans the “transmitters of ancient Babylonian algebra to the
Greek world” (MtAnn 1940-1941, 160; cf. MtAnn 1947-1949, 132). Neugebauer is
hesitant: “My answer to this question cannot be proved by documentary evidence”
(1ixSc 147); so is G. Junge, Osiris 8 (1948) 316-321). A more direct kind of appropriation
of Babylonian arithmetic is represented by the Pythagorean number series (above, ch.
VI 2, esp. nn. 9 and 43).
ὅδ Above, ch. Π 4, nn. οὐ ff.
8 Above, ch. III 1, nn. 28 β΄ Burkert, Philologus 1961.
* Timaeus FGrHist 566F14 -- D.L. 8.54 (AoyoxAomia); Neanthes FCrHist 84F26 =
D.L. 8.55.
38 Above, ch. III 1, n. 28.
39 Above, ch. IV 1, ἢ. 37.
454
4. Pythagorcan Geometry and Mathematical Secrets
called Simos" and Plato! ‘The oldest authorities are ‘Timaeus of
Tauromenium and Duris of Samos, who both seem to show a local
patriotic partiality to Pythagoras. It is significant that, in the case of
Oenopides, Eudemus, the carlier witness, has nothing to aay of pla-
giarism. A more famous and morc important story is that of the mathe-
matical treason of Hippasus, bringing with it the impressive anecdote
that he was drowned at sea, as punishment; and this same story is
bound up with the discovery of the irrational.
The discovery of the problem of the irrational in geometry, and the
development of the ability to cope with it, is a fundamental accom-
plishment of Greek mathematics which holds a lasting fascination for
modern historians of science. The tradition of secrecy, betrayal, and
divine punishment provided the occasion for the reconstruction of a
veritable melodrama in intellectual history. The realization that certain
geometrical magnitudes are not expressible in terms of whole numbers
is thought of as “une véritable scandale logique,” bound to shake the
very foundations of the Pythagorean doctrine, which maintained that
“everything is number’; for to the Greeks, number and irrationality
are mutually exclusive.*? Thus one comes to speak of a Grundlagen-
krisis—a crossroads or dilemma as to the very foundations of Greek
and Pythagorean mathematics in the fifth century? —and to see in
the tradition about the death of the “traitor” a reflection of the shock
and despair that this discovery must have brought: “ O that the irrational
40 Duris FGrHist 76F23 = Por. VP 3 = DK §6.2: Pythagoras’ son Arimnestus is said
to have set up in the sanctuary of Hera (on Samos) a dedicatory monument with the
epigram ἫΝ
Πυθαγόρεω φίλος vids ᾿Αρίμνηστός μ᾽ ἀνέθηκε,
πολλὰς ἐξευρὼν εἰνὶ λόγοις σοφίας. ! ne
τοῦτο δ᾽ ἀνελόντα Lipoy τὸν ἁρμονικὸν καὶ τὸν κανόνα σφετερισάμενον ἐξενεγκεῖν ὡς
ἴδιον. εἶναι μὲν οὖν ἑπτὰ τὰς ἀναγεγραμμένας σοφίας, διὰ δὲ τὴν μίαν, ἣν Σῖμος ὑφείλετο,
συναφανισθῆναι καὶ τὰς ἄλλας τὰς ἐν τῷ ἀναθήματι γεγραμμένας. Diels (DK I 445 1)
thinks the σοφίαι referred to are the μεσότητες, but the seventh of them was not discovere
till after Eratosthenes (above, ch. VI 2, n. 92). The word κανών suggests rather the calculation
of a scale (as Wilamowitz, Platon II 94). There were therefore, aside from the Pythagorean
musicologists, other dppovxoi in competition with them.
41 Above, ch. III 1, n. 28. | “ΝΕ [
# Tannery, MSc I 268, Géom. 98, HScH 259; Arist. Met. τοζιας: 6 γὰρ ἀριθμὸς
σύμμετρος. In Diophantus the expression occurs that the number sought γίνεται οὐ
pyres (e.g. 4.10); see esp. Hasse~Scholz 65f; below, n. 71. ; .
48 Hasse-Scholz; cf. Heath, Math. I, 155; Brunschvicg, Etapes 45ff, Le réle du pythagorisme
dans l’évolution des idées (Paris, 1937) 21ff (chapter heading : “‘ Une découverte scandaleuse”);
von Fritz, AnnMath 1945, 244f; Sarton, Hist. 283f; Becker, MD 13f, 7uff : Heller, AbhBln
1958, 11; Junge, C&M 1958, $3f (though at 67f, Junge rejects the idea of Soe
aus schlechtem Gewissen,” and maintains, correctly, that the Pythagoreans were apparently
not upset by the matter).
4545
had never been discovered!" Hat had the painful fact of its existence
teally been held as a carefully μυμε οι arcanum iimperit 245
Av important prop for thiv theory of a Grundlagenkrisis, and also a
useful chronological point of reference, was the interpretation of the
polemics of Zeno of Elea as relevant to the history of mathematics.
This was inaugurated by Helmut Hasse and Heinrich Scholz, who
argued that his critique was directed specifically against some “un-
clean” (unsauber) mathematics of infinitesimals, by means of which the
Pythagorcans supposedly attempted to escape the consequences of
irrationality. If this is correct, the discovery of the irrational must
have taken place before 460 3.c., which would fit in well with the
conjectural dating of Hippasus.4”
More recent interpretation of Eleatic philosophy does not confirm
the attempt to place Zeno in the history of mathematics.48 His argu-
ments are sometimes given a specific mathematical purport*® which is
in no way suggested by his words, and which antiquity did not find in
them, assigning him, as it did, to the ranks of the φυσικοί rather than
the μαθηματικοί. 50 Even when one distills out the purely mathematical
content of his arguments, and does find the concept of the infinitely
small and of the infinite series, still these series converge in a rational
value, not in irrational proportions.*! Certain as it is that Zeno’s
arguments are relevant to mathematics and even contributed in a
certain way to the development of Greek mathematics,® still, from
the historical point of view ontology is prior to mathematics.®3
*4 Junge, CEM 1958, $3; cf. Heath, Becker, von Fritz AnnMath 1945, 260.
“Im schlimmsten Sinn unsaubere Methoden,”’ Hasse-Scholz το.
46 Hasse-Scholz 8ff, and J. Zafiropulo, L’Ecole éléate (Paris, 1950) 178ff. On the other
hand, Mondolfo (Inf. 238ff )and Junge (C&M 1958, 54ff ) think that the fact of irrationality
was not concealed but openly admitted, and that this stimulated the development of
Zeno’s arguments.—Philolaus A26 aims to avoid the irrational, but this is not “unclean”
mathematics; it is not mathematics at all.
4“ Von Fritz, AnnMath 1945, 245ff; cf. above, ch. II 5, nn. 69-76.
“8G. E. L. Owen, “Zeno and the Mathematicians,”’ Proc. of the Arist. Soc. 58 (1957-
$8) 199-222, calls Tannery’s thesis “an obstructive myth” (212); also see G. Vlastos,
Eneyc. of Philos. VIM (New York, 1967) 376-377.
* εἰ πολλά ἐστιν (fr. 1, fr. 3) is supposed to mean “‘if it is legitimate to regard a line as
an aggregate of infinitely many, infinitely small ‘atomic’ lines” (Hasse-Scholz 10);
sce above, ch. III 3. .
* Stressed by Heidel, AJP 1940, 25 n. 54; Hasse and Scholz maintain (12) that Zeno
belongs in the history of mathematics, rather than in that of Sophism.
δ᾽ Emphasized by Becker, Gnomon 27 (1955) 267.
*2 Zeno, to some extent, obstructed the development of the calculus of infinitesimals
among the Greeks; and Archimedes uses his ἔφοδος, which is equivalent to integration,
only as a heuristic method, which must be followed by a rigorous proof (see, e.g., van
der Waerden, SA 224f).
δδ Cf. above, ch. VI 1, n. 138: III 3, n. 44.
456
Fr, = ,Ἣέ͵ ΠΡ ῬΑ ΡΣ Oe LT νου" ἃ aa
Furthermore, ti not attested in any ancient source that ΕΠ βάσεις
discovered the irrational, or divulged this knowledge; side by side
stand diverse accounts, mutually exclusive, of the secrecy and the
publication of Pythagorean mathematics. According to the version of
the Pythagorean mathematici, obviously recorded by Aristotle,
Hippasus was the first to “publish and construct” the “sphere of the
twelve pentagons,” that is, the dodecahedron; he was drowned at sea
as a punishment for this offense, but got the reputation of being the
discoverer. According to the contrary version of the acusmatici,
Hippasus was clearly the man who, by his discovery, inaugurated a
new trend in Pythagoreanism, different from the original. There is no
direct mention, here, of the irrational, the “betrayal” of which appears
in a somewhat different account. Plutarch, who is our oldest witness
for this, speaks of the secrecy, and the prohibition of putting doctrines
down in writing, in the Pythagorean group: “And when their treat-
ment of the abstruse and mysterious processes of geometry had been
divulged to a certain unworthy person, they said the gods threatened
to punish such lawlessness and impiety with some signal and widespread
calamity.”’5> We cannot equate this episode with that of Hippasus; the
latter was a Pythagorean, so that his initiation into the “difficult and
secret procedures’”’ was therefore not any kind of “‘divulgement,”
and his death was not a κοινὸν κακόν.
The next witness is Pappus:*
This science (or knowledge) had its origin in the sect (or school) of
Pythagoras, but underwent an important development at the hands
of the Athenian, Theaetetus ... Indeed the sect (or school) of
Pythagoras was so affected by its reverence for these things that a
saying became current in it, namely, that he who first disclosed the
knowledge of surds or irrationals and spread it abroad among the
common herd perished by drowning; which is most probably a
parable by which they sought to express their conviction that firstly,
it is better to conceal (or veil) every surd, or irrational, or incon-
ceivable in the universe, and, secondly, that the soul which by error
or heedlessness discovers or reveals anything of this nature which is
*“ Cf. above, ch. II 5. Iam. Comm. math. sc. p. 77.18 = Iam. VP 88: περὶ δ᾽ Ἱππάσου
λέγουσιν, ὡς ἦν μὲν τῶν Πυθαγορείων, διὰ δὲ τὸ ἐξενεγκεῖν καὶ γράψασθαι πρώτως
σφαῖραν τὴν ἐκ τῶν δώδεκα πενταγώνων ἀπώλετο κατὰ θάλατταν ὡς ἀσεβήσας, δόξαν δὲ
λάβοι ὡς εὑρών, εἶναι δὲ πάντα ἐκείνου τοῦ ἀνδρός...
55 Plut. Numa 22, tr. Perrin.
S11, p. 63; 1 2, p. 64 Junge~Thomson (cf. above, ch. VI 2, ἢ. 82). Pappus is dated,
approximately, 300 A.D.
457
vee me PA se oe SO eee eee as eas SS Se τῷ δ
initorin this world, wanders (thereafter) hither and thither on the
sca of non-identity (ic. lacking, all similarity of quality or accident),
mumersed in the stream of the conung-to-be and passing~away,
where there is no standard of measurement. This was the considera-
tion which Pythagorcany and the Athenian Stranger held to be an
incentive to particular care and concern for these things. . .
The scholia to Euclid have preserved an abbreviated version of the
Greek text,” and Iamblichus, too, knows the tradition Pappus is
following. But he also has another, according to which the traitor
was only symbolically killed—a tomb was erected with his name.
Iamblichus scts the three versions side by side—the symbolic “death”
of the betrayer of irrationality, the drowning of the man who revealed
the dodecahedron, and the drowning of the one who divulged the
fact of irrationality.58 In addition, Iamblichus has a different story about
how Pythagorean geometry became known, without any quarrel or
catastrophe: permission was given an impoverished Pythagorean to
earn a living by giving lessons in geometry. It is conjectured that this
version was originally related to Hippocrates of Chios.®®
δ᾽ Schol. Eucl. 417.12ff: τῶν γὰρ Πυθαγορείων λόγος τὸν πρῶτον τὴν περὶ τούτου
θεωρίαν εἰς τοὐμφανὲς ἐξαγαγόντα ναυαγίῳ περιπεσεῖν, καὶ ἴσως ἠνίττοντο, ὅτι πᾶν τὸ
ἄλογον ἐν τῷ παντὶ καὶ ἄλογον καὶ ἀνείδεον κρύπτεσθαι φιλεῖ, καὶ εἴ τις ἂν ψυχὴ ἐπιδράμοι
τῷ τοιούτῳ εἴδει τῆς ζωῆς πρόχειρον καὶ φανερὸν τοῦτο ποιήσηται, εἰς τὸν τῆς γενέσεως
ὑποφέρεται πόντον καὶ τοῖς ἀστάτοις ταύτης κλύζεται ῥεύμασιν. The double ἄλογον
of the manuscripts must be a corruption; but there were 3 elements in Pappus’ sentence,
too. Should we perhaps read ἄπορον as in Plutarch ¢—Probably we should read ἐπιδραμοῦσα
for ἐπιδράμοι. Heiberg’s conjecture was εἴ Tis. . , ψυχῇ ἐπιδραμὼν... .; but in Pappus the
subject is “soul.”.—An interpolation in Elias (In Arist. Cat. 12 4.12; CAG XVIII 1) reads:
ws καί τις τῶν Πυθαγορείων ἐκδοὺς τὸ μονόβιβλον τὸ περὶ τῶν ἀλόγων γραμμάτων
ναυαγίῳ περιέπεσεν, ὡς τὰ ἀπόρρητα ἐξορχησάμενος. The specific detail about the
μονόβιβλον probably stems from confusion with the ps.—Aristotelian De lineis insecabilibus
(Περὶ ἀτόμων γραμμῶν).
*®Tam. VP 246f, evidence of their ἐχερρημοσύνη: τὸν γοῦν πρῶτον ἐκφάναντα (sic)
τὴν τῆς συμμετρίας καὶ ἀσυμμετρίας φύσιν τοῖς ἀναξίοις μετέχειν τῶν λόγων οὕτως φασὶν
ἀποστυγηθῆναι, ὡς... καὶ τάφον αὐτοῦ κατασκευασθῆναι... (247) οἱ δέ φασι καὶ τὸ
δαιμόνιον νεμεσῆσαι τοῖς ἐξώφορα τὰ Πυθαγόρου ποιησαμένοις: φθαρῆναι γὰρ ὡς
ἀσεβήσαντα ἐν θαλάσσῃ τὸν δηλώσαντα τὴν τοῦ εἰκοσαγώνου σύστασιν" (this is repetition
of the report about Hippasus, Iam. VP 88; above, ἢ. 54; cf. Sachs 83, following Rohde,
Q 168; thus the specification τοῦτο δ᾽ ἦν δωδεκάεδρον, ἕν τῶν πέντε λεγομένων στερεῶν
σχημάτων, εἰς σφαῖραν ἐντείνεσθαι is the work of Iamblichus) ἔνιοι δὲ τὸν περὶ τῆς
ἀλογίας καὶ τῆς ἀσυμμετρίας ἐξειπόντα τοῦτο παθεῖν ἔλεξαν.
°° Tam. VP89= Comm. math.sc. p.78.1ff: λέγουσι δὲ of IT. υθαγόρειοι ἐξενηνέχθαι γεωμετρίαν
οὕτως. ἀποβαλεῖν τινα τὴν οὐσίαν τῶν Πυθαγορείων, ὡς δὲ τοῦτ᾽ ἠτύχησε, δοθῆναι αὐτῷ
χρηματίσασθαι ἀπὸ γεωμετρίας. This stands in the context of the material taken from
Aristotle, but is obviously an interpolation made by some compiler (as is the following
sentence; above, ch. VI 1, n. 43); the Aristotelian passage is concerned with the difference
between mathematici and acusmatici, but here one hears simply of ‘‘Pythagoreans.” To
the report of Hippasus’ ἐξενγκεῖν is added a rival report of the ἐξενηνέχθαι of geometry
in general, which takes no note of Hippasus. The preceding context has another sentence
458
4. Pythagorean Geometry and Mathematteal Secrets
Can we reconcile these various versions, ina spirit of COMproMise,
with the thesis that Hippasus discovered or made known the dode-
cahedron “as well as” (“und cbenso”’) the irrational; or, as Kurt von
Fritz ingeniously suggested, that Hippasus discovered the irrational
in the case of the dodecahedron, that is to say, of the regular pentagon?"
The fact of irrationality, of infinite reciprocal subtraction, is casily
seen in the case of the “Golden Section”; but there is no hint, in the
tradition, that this was the point of departure for the discovery of the
irrational.® There are two branches of the story of the traitor: on one
hand Hippasus, the dodecahedron, and drowning at sea, and on the
other the discovery of irrationality but no name mentioned, and
various reports about the nature of the penalty. The explanation of the
offender’s death as symbolical is found in the report of Nicomachus,
who referred to the letter of Lysis;5* and the motif of drowning, of
which Plutarch makes no mention, may also be an accretion to the
story of the betrayal of irrationality.
All the accounts involving some kind of mathematical secret and its
(Comm. math. sc. 77.24) with the key word ἐκφέρειν: ἐπέδωκε δὲ τὰ μαθήματα, ἐπεὶ
ἐξηνέχθησαν, δισσοὶ mpodyovre μάλιστα, Θεόδωρός τε ὁ Κυρηναῖος καὶ “ἱπποκράτης
ὁ Χῖος. Τῆς language has peculiarities (plural verb with neuter plural subject; δισσοί = δύο,
an Ionic and poetic usage, in any case not Aristotelian; and an impossible participle in
the dual). The content corresponds to Procl. In Eucl. 66.4 = Eudemus fr. 133, from whose
source Iamblichus probably derived the interpolated passage. (This was noticed by
Tannery, MSc VII 112, and von Fritz, AnnMath 1945, 245, though the latter wrongly
attributes to Eudemus the preceding report about Hippasus, which is not closely connected
with the sentence in question. See also Heller, AbhBln 1958, 7).—Aristotle reports that
Hippocrates of Chios lost his property (DK 42.2); and this seems to be the source of this
version (Sachs 120.1).—Tannery, MSc VII 115ff, Géom. 84f, HScH 124, takes the report
seriously, and suggests that after the catastrophe of the Pythagorean rule, under the
pressure of material need, the Pythagoreans published the supposed Tradition suivant
Pythagore. (See above, ch. VI 1, n. 44.)
6° Becker, MD σι.
®? Von Fritz, AnnMath 1945; following him, Heller, AbhBli 1958, off; Junge, CGM
1958, 42.
ὡ To my knowledge, the connection of the regular pentagon with the irrational is
never emphasized in the tradition. The paradigmatic example for the irrational is always
the diagonal of the square (below, nn. 75, 81); and von Fritz, in an earlier article (RE
V A 1813), assumed that the diagonal was ‘“‘zweifellos” the point of departure for the
discovery.
68 Jam. VP 74f (from Nicomachus according to Rohde, Q 138; at lam. VP 253 = Por.
VP 58, Nicomachus cites the letter of Lysis) and Clem. Al. Strom. 5.57 (from Nicomachus?),
citing the letter of Lysis, which was addressed to a certain Hipparchus (Hippasus D.L.
8.42; but Clement, Iamblichus, and the MSS of the collection of letters all have Hippar-
chus). The latter had the temerity to δαμοσίᾳ φιλοσοφέν, and is given the threat, if he
does not mend his ways, τέθνακάς μοι. The complete text is in Hercher, Epistologr. gr.
p. 601-603; Thesleff, Texts 111-114. See Burkert, Philologus 1961, 1ff, where it is con-
jectured that the letter was forged in the 2nd half of the 3rd century B.c., as an intro-
duction to the Hypomnemata.
459
. PYTHAGOREAN NUMBRER THEORY
betrayal share the same inherent improba bility, which was perceived by
Tannery. Fame and profit can only resule from an invention if it is
welcomed by an expectant public, so that a mathematical discovery is
only of interest in mathematical circles—mathematica mathematics
scribuntur. Nobody who is not already schooled in mathematical
logic is going to be deeply impressed by the fact of irrationality. But
if mathematical logic was already in existence, the soil was prepared for
further development; and in mathematics it is especially easy for differ-
ent students to come to identical results. If mathematics had been an
exclusive possession of the Pythagoreans, its betrayal would have been
meaningless; but if the problems and methods of deductive mathe-
matics were already present outside the Pythagorean circle—a fact
not to be doubted—then this ἐκφέρειν could at most hasten a process
that was already going on.
This dilemma, however, does not apply in the case of the dodeca-
hedron. As in the case of musical discoveries (ch. V 1), the tradition
about Hippasus, though surrounded by legend, makes sense. In the
background of the mathematical problem of the dodecahedron there
stands the dodecahedron as a cult object. Numerous dodecahedra
made of bronze have been found in Gaul and thereabouts; and one
made of stone has been found in northern Italy, dating back to pre-
historic times. Their significance and use is unclear: the best con-
jecture seems to be that they were a kind of dice, used for oracular or
mantic purposes. In Plato’s Timaeus the dodecahedron appears un-
expectedly as the image of the whole (55c); it is widely supposed that
the Pythagorean tradition was in his mind, the one that presupposed
the Hippasus story and was not without relation to the Italo-Gallic
region.*® The dodecahedron may well have been important as a
σύμβολον in the Pythagorean school, like the pentagram; Hippasus’
offense was in analyzing the sacred object, publicly, by mathematical
means. Whether he actually made a mathematical construction of it is
*¢ Tannery, Géom. 82f. In spite of this, he accepted the tradition of secrecy, because of
the supposed “primitiveness” of the mathematics of Oenopides (n. 31).
85 F. Lindemann, “Zur Geschichte der Polyeder und der Zahlzeichen,” SBMii, math.-
ph. KI, 26 (1897) 625-768, described 28 dodecahedra, including one of steatite from
Monte Loffa, northern Italy, dated 9th to 6th century B.C. (cited by Sachs 83f). On the
(more than 40) Celtic dodecahedra, see J. de Saint-Venant, Dodécaédres perlés en bronze
creux ajouré de ’époque gallo-romaine (Nevers, 1907); L. Saint-Michel, Bull. de Ass. C.
Budé 3rd ser. 4 (1951) 92-116; and W. Deonna, Bull. de Ass. Pro Aventico 16 (1954)
19-89, who makes it seem probable that they were used in a “jeu divinatoire.” Saint-
Michel and Deonna ascribe the Celtic dodecahedra to Pythagorean influences, but the
prehistoric example would rather indicate that here, again, Pythagoras was dependent on
older material.
6 Cf,eg., Taylor, Tim. 377-378.
460
Pythagorean Geometry and Mathematical Secrets
uncertain,”? and itis all the more doubtful whether, in this Process,
he stumbled onto the problems of the irrational.
It is striking that in the accounts of the “betrayal of the secret of the
irrational” the oldest, that of Plutarch, is the least precise, that Pappus
speaks only of a legend which arose as a result of the discovery, and
that all the reports stress the deep significance rather than the facts of
the matter.** In Plutarch it is clear that the word ἄρρητος, set in quotation
marks, as it were, by λεγόμεναι, is to be understood in a double sense.
The “ineffable because irrational” is at the same time the “ unspeakable
because secret.” Carefully guarded secret doctrines (dppyra)—which
are dangerous to the uninitiate—played an important part, in Plu-
tarch’s day, in all the mysteries and similar organizations.6* Thus
Pythagorean philosophy was, as Nicomachus puts it, ἄρρητος ἐν
τοῖς στήθεσι διαφυλαχθεῖσα.70 Not that it was arbitrarily “forbidden”
—arroppyros—but it was “unsayable.”’ The fascination of the ἄρρητον
lies in the pretense to indicate the fundamental limitations of human
expression, which are at the same time transcended by the initiate.
It is no wonder that this fascination was felt in the realm of mathe-
matics as well; Athenagoras reports of the Pythagorean Lysis the
definition, ἀριθμὸν ἄρρητον ὁρίζεται τόν θεόν.71 This exciting double
sense of the word dppyros is what makes the story of the discovery and
betrayal of the irrational an exemplum for Plutarch, and even more for
Pappus, who is probably following some Platonic source. When we
see that a name is not mentioned, and that the details vary from one
version to another, it is tempting to think that it was precisely the
®’ Sachs 82: “‘That the mathematical construction of the dodecahedron took place
before that of the octahedron—when all the solids were known-—is unthinkable," She
translates γράψασθαι “sich zeichnen” (83), and supposes that the Pythagoreans knew
the dodecahedron only in an empirical way. The expression δωδεκάσκυτοι σφαῖραι
in Ρ], Phd. r10b (cf. Plut. Quaest. Plat. $.1.1003d), shows that (as today) balls for children
were made in the shape of dodecahedra, of 12 pieces of leather; this makes it seem unlikely
that the dodecahedron was first made known, outside of Pythagorean circles, by
Hippasus. .
°8 Pappus expressly cites the “Athenian stranger” from PI. Leg. 819d, and in general
his exposition is strongly influenced by Platonism. 7
9 ἄρρητα ἱερά as early as Herodotus 5.83, 6.135, Aristoph. Nub. 302, Eur. 11 41, Bacch.
472, etc. On the ἄρρητον in Plato, see Friedlander, Platon 1 63ff = 1 soff Eng. tr. From
late antiquity, see for example Plut. De Is. et Os. 25: ὅσα τε μυστικοῖς ἱεροῖς
περικαλυπτόμενα καὶ τελεταῖς ἄρρητα διασῴζεται; De an. Procr. 4.1013¢ (the argument
that the world was created, and therefore not eternal): ὥς τι δεινὸν καὶ ἄρρητον οἰόμενοι
δεῖν περικαλύπτειν καὶ ἀρνεῖσθαι, Hippol. Ref. 1 pracf. 1 (on the Gnostics) ἀποκρύπτειν
τὰ ἄρρητα ἑαυτῶν μυστήρια...
70 Nicom. ap. Por. VP §7 = Iam. VP 2§2. | |
1 Athenag. Suppl. 6 =- DK 46.4 =. p. 114.15 Thesleff. Of course this is as spurious as
the other Pythagorean quotations in Athenagoras, bue still apparently the carliest extant
reference for the expression “irrational number" (see above, n. 42).
461
EEE EEO EE NA δ ἀν δ Δ Ὺ δ WEE esa δ TINIAN ET
ambiguity of the word ἄρρητον which provided the germ of the
tradition, or at least that thiv was an influential factor in its develop-
ment, One had the story of Hippasus as told by Aristotle, the Lysis
letter with its threat of symbolic death, and, in general, the careful
secrecy of the Pythagorean group. Pythagorean lore was dppyros
like that of the mysteries,™ and the penalty of betrayal was death.
Pythagorean lore was mathematical in nature, and in geometry ἄρρητα
play a role which is mysterious to a layman; how easy a step to equate
the two senses of dppyros and create the story of the secret of the irra-
tional and its betrayal! The authority of Eudemus fails us twice, in
Pappus and in Proclus;”8 but it may underlie the scholia on Euclid.74
Thus it is possible that he mentioned the fact of the Pythagoreans’
discovery, giving the σὰς for the development of the legend. But the
involvement of Hippasus is scarcely better attested than that of Zeno,
so that another apparent chronological foothold is lost.
The deep significance of the discovery, so dramatically expressed in
the catchword Grundlagenkrisis, is not attested in the sources. “N owhere
in the many passages about the irrational in Plato and Aristotle can we
detect any reference to a scandal, though it would surely still have been
known in their day.”?5 In addition, the inherent connection of the
problem of the irrational with Pythagorean speculation and philosophy,
which some have supposed they saw, is doubtful. The so-called
Pythagorean theorem, when applied to a square, would necessarily
lead to the irrational /2; but Babylonians, Indians, and Chinese
knew the theorem without knowing about the irrational. Side numbers
The comparison with the Eleusinian mysteries is drawn in the letter of Lysis, p.
112.3ff Thesleff = Hercher, Epistolog. gr. p. 602.4ff = Iam. VP 75.
ὁ Pappus cites Eudemus for the achievement of Theaetetus (above, ch. VI 2, n. 82),
but not in connection with the Pythagoreans (above, n. 56). After the introductory
sentence on Pythagoras follows a citation of Plato (Tht. 143e); then Apollonius of Perga
is mentioned, and Eudemus makes his appearance only after a second Platonic citation
(Tht. 1470), so that the entire Passage cannot be ascribed to him. See above, n. 68.
* Schol. Eucl. pp. 415. 7~416.13, characteristically speaking of Πυθαγόρειοι; cf. the
introductory sentence of Pappus (above, n. 56): “This science had its origin in the sect
of Pythagoras, but underwent an important development at the hands of the Athenian
Theactetus.” An expedient of practical geometry was to speak of two different
“measures,” e.g. of the side and the diagonal of the square, Schol. Eucl. p. 416. 8-10,
Arist. Met. 1os3a14ff (misunderstood by Ross, Met. II 283; not explained by Heath,
Aristotle 218), Pl. Parm. 140b-—c.
** Reidemeister 30. Van der Waerden had opposed the theory of a Grundlagenkrisis
(MtAun 1940-1941, 155): “not a philosophical problem, but one that arose within the
development of mathematics itself”. See also von Fritz, ABG 1955, 84f. The expression
ἄλογοι ὥσπερ γραμμαί seems almost proverbial as early as Plato’s Republic (534d). It is an
important μάθημα, Leg. 819d; cf. Arist. Met. 983aI19ff: οὐθὲν γὰρ ἂν οὕτως θαυμάσειεν
ἀνὴρ γεωμετρικὸς ὡς εἰ γένοιτο ἡ διάμετρος μετρητή.
462
3 Pythagorean Geometry and Mathematical Secrets
and diagonal numbers belong in the same context but do they
represent a way of understanding or of avoiding the irrational?” ‘The
arithmetical proof of the irrationality of /2 is regarded as carly,
but it does not emerge naturally as a consequence of the Pythagorean
pebble games.”” Music theory advances as far as the problems οἱ the
irrational, but stops there and declares them nonexistent.”8 The irrational
belongs to the domain, not of arithmetic, but of geometry.
For the Pythagoreans who were concerned with the number theory
described by Aristotle, and for the cosmology summarized in the phrase
“everything is number,” the irrational has obviously no importance.
Eurytus, who cannot be dated earlier than about a generation before
Archytas,’® tried to determine “the number” of specific objects,
oblivious that the discovery of the irrational had long before cut the
ground from beneath the Pythagorean theory of numbers. What is
more, Aristotle, though he knew about it, did not use the fact of
irrationality as an argument against the Pythagorean doctrines he
criticizes, while the book On Indivisible Lines does use irrationality
against the “atomic lines” of Xenocrates.8° Clearly Pythagorean
number theory and deductive mathematics lie on two different planes;
“all things are number” never means “all magnitudes are commen-
surable.” In Pythagorean number theory the relations of existing
things are interpreted, and the “‘nonexistent”’ is left out of account.
In attempting to date the discovery of the irrational, we have the
known fact that Theodorus of Cyrene proved the irrationality of
square roots in the cases between 4/3 and 4/17, so that the proof for
./2 was known before his time.8! Democritus seems to have been
76 Above, ch. VI 2, nn. 14-18.
” Above, ch. VI 2, n. §4.
78 Above, ch. V 1, n. 4. ΝΕ
19 DK 45. How far these Pythagoreans were from being true mathematicians is shown
by the fact that their ‘‘units,”’ according to Aristotle, ‘“‘had magnitude, while even
Protagoras presupposes a definition of the point, among geometers, as without extension.
80 Ps.-Arist. Lin. ins. 969b33ff. Junge pointed to Eurytus, in this connection, as early
Symb. Joach. 230ff); cf. C&M 1958, §7ff.
᾿ o PL i "3; As ᾿ ue method of proof Theodorus may have used, there has been
a long discussion, not yet concluded, among modern scholars. Whereas Vogt (Bibl.
Math. 1909-1910, 1018) and Junge (C&M 1958, 42ff) reconstructed a geometrical proof
making use of reciprocal subtraction, and van der Waerden (SA 142ff ) advocated an
algebraic-mathematical method; von Fritz (RE V A 181 5ff), Cherniss (Rev. of Metaph. 4
[1951] 411f), and A. Wasserstein (‘‘ Theaetetus and the History of the Theory of Numbers, :
CQ 52 [1958] 165-179) favored an apagogic method analogous to the numerical proo
of the irrationality of 4/2 (above, ch. VI 2, n. 47). The irrationality of 4/2 can very
easily be proved for the square by the method of reciprocal subtraction (van der Waerden,
MtAnn 1947-49, 689f, SA 127; Becker, MD 73f). The resulting διάγραμμα is identical
to that of the “side” and “diagonal”? numbers, above, ch. VI 2, n. 16.
463
concerned with the ργοθίοην ΜῈ but how far back into the fifth century
we should date the divcovery ts. an open question. If we try to determine
the position of Hippocrates of Chios, the results are as ambivalent as
those relating to the problem of the application of arcas (above, n. 25).
He obviously has no trouble constructing a line with a length equal to
J%,83 and #2 is also a geometrical problem for him. One finds the
square root by constructing the mean proportional; “ geometrical
algebra” is on the way. But, on the other hand, Hippocrates uses a
definition of the equality of ratios which is not applicable to irrational
magnitudes. The problem in the theory of proportion which this
raises was solved only by Eudoxus;® and a passage in the Academi-
corum index seems to allude to a forward step between Hippocrates
and Eudoxus: in Plato’s time, it says, (ἦλθεν εἰς κορυφὴν)... τὰ περὶ
ἰτοὺς ἀρι]θμοὺς προβλήματα, τῶν περὶ ΕἸ ὔδοϊξον μετάστησάντων τὸν
ald’ “Ιπ]πο[κράτους ἀρχαιομόν. 85
*? He has a title (Β11ρ) περὶ ἀλόγων γραμμῶν καὶ ναστῶν a’ β΄ (ναστόν = πυκνόν, ὄν).
Vogt (Bibl. Math. 1909-10, 144ff) contests the reference of this to the problem of the
irrational, since he wishes to show that the word ἄλογος in the sense “irrational” is
post-Platonic, overlooking Pl. Rep. 534d. One can only guess how Democritus solved
the problem: for a spatial atomism are 5. Luria (QSt 2 [1932] 106-185) and Heath,
Aristotle σοῦ, for the distinction of geometrical space from the atoms (following Simpl.
Phys. 82.1, Schol. Arist. 469b14) are R. Philippson, Hermes 64 (1929) 177ff, and Michel
675 n. I.
8 γραμμὴ ἡμιολία δυνάμει, Eudemus fr. 140.
δ: Hippocrates takes his departure from the proposition that similar segments are
related as the squares of their bases. The proof is that circles are related as the squares of
their diameters: ὡς γὰρ of κύκλοι πρὸς ἀλλήλους ἔχουσιν, οὕτως καὶ τὰ ὅμοια τμήματα'
ὅμοια γὰρ τμήματά ἐστι τὰ τὸ αὐτὸ μέρος ὄντα τοῦ κύκλου (Simpl. Phys. 61.11f€; cf.
Eucl. 7 def. 21; the exact proof was first offered by Eudoxus; for a conjecture about
Hippocrates’ proof, see Becker, Gnomon 27 [195 5] 267 n. 4). This sentence is regarded by
Tannery (MSc I 3006), Diels (ed. of Simpl. in CAG), and Becker (QSt 3, 411-410) as
an addition of Simplicius, and Wehrli omits it. The ascription to Eudemus was defended
by Rudio (Bibl. Math. 3 [1902] 1-62—where he weakens his argument with the claim that
τμῆμα first means sector, and then in the following sentence segment; refuted by Tannery,
MSc ΠῚ 119-130 = Bibl. Math. 3 [1902] 342-349). See also the discussion of Heath, Math.
I 187ff. The citation from Euclid inserted by Simplicius ends before ὡς yap... The word
ἐδείκνυεν (p. 61.8) is followed by the indication of the course of proof, which is followed,
as a consequence (διὸ καὶ, p. 61.14; Wehrli and Diels have the citation of Eudemus take
up with καὶ, without διό), by the sentence about the angles in similar segments. There-
fore, the proposition using the definition of proportionality, which is valid only for
numbers, not for geometrical magnitudes (Euclid 7 instead of the general theory of
proportion in book 4) may be regarded as evidence from Eudemus about Hippocrates.
After Zeuthen, van der Waerden called attention to the passage, MtAnn 1940-1941,
1$7f; cf. von Fritz, ABG 1959, 6off, 6of.
*° See Becker, MD το; van der Waerden, SA 187ff; Heath, Math. I 32ff. A “pre-
Eudoxan” theory of proportion, which was applicable to irrational magnitudes, and
which operates with reciprocal subtraction, was reconstructed by Becker (QSt 2 [1932]
311-333; cf. Grdl. 79ff, MD 103f: von Fritz, ABG 1955, 92; van der Waerden, MtAnn
1947-1948, 688f, SA 177ff; Heath is skeptical, Aristotle 80ff).
* Page 16 Mekler. A perfectly certain restoration is hardly possible.
464
The conquest of the miational in Greek mathemation, and alonp with
it the development of “yeometrical algebra,” clearly was a gradual
process. ‘There were beginnings before Hippocrates of Chios, but the
decisive breakthrough was later. The Pythagorcans to whom Buenas
attributed the application of areas and perhaps also the “discovery
of the irrational may well belong to this later period, Whether it wan
the same mathematician who first grasped the idea of the ἀσύμμετρον
and “demonstrated” it by alternate subtraction of side and diagonal of a
square, who exploited the superiority of geometry to algebra in the
development of the application of areas, and who perceived the
difficulty that the ἄλογοι γραμμαί do not fit into the calculation (ratio,
Adyos)—or whether several men worked, one after the other, on these
problems—we can no more answer these questions than we can give
their names. But in any case, the beginnings of the development that
is brought to a climax by Eudoxus are not far from Theodorus of
Cyrene.®’ This strictly mathematical development had little relation
to the cosmological number theory of men like Philolaus and Eurytus.
Pythagoreans made significant contributions to the development of
Greek geometry, but the thesis of the Pythagorean foundation of
Greek geometry cannot stand, any more than the legend of a great
mathematics held secret.
4. NUMBER AND COSMOS
Aristotle represents the Pythagoreans’ number theory as arising from
their study of exact mathematics,! and this relationship may well have
seemed obvious, or even necessary. To be sure, there is no evidence,
and no way of showing by philological means, that the Pythagorcans
87 Frank concludes from a passage in Plato’s Laws (819d) that in his time the problem was
quite new, so that before 400 it was ‘completely unknown among those most keenly
interested in scientific matters”’ (228): it was brought to Athens by Theodorus in 399 B.C.
But since at least half a century elapsed between 400 and the composition of the Laws,
the word ὀψέ in 819d cannot be pressed (cf. von Fritz, AnnMath 1945, 243 ἢ. 6). Vogt,
Bibl. Math. 1909-1910, dated the decisive discovery about 410, van der Waerden (MtAnn
1947-1949, 153) before 420, and von Fritz, thinking of Hippasus, about 450 (ABG
1955, 84 n. 159). A simple point of departure for the discovery was provided by the
traditional (partly Babylonian) approximative values for the diag onal of a square. When
the Greeks endeavored to find an exact result to replace the ap proximations, they dis-
covered the fact of incommensurability, in the same way that they recognized the squaring
of the circle as a problem.
1 Above, ch. VI 1, n. 68.
465
οἱ Oe _ ἂν ὙΥ Be gens τοῦτ’ 70 Ὸ ee ete τ ame Ne ey. ee Ἢ ΒΘ ΟΝ ΟΣ ἸΌΝ ΟΝ ee a = =
were the originators of mathematics as a deductive science; and it is
impossible to discern any relationship between the doctrine of trans-
migration, which is reliably attested for them, and the science of
mathematics. Number does, however, dominate the Pythagorcans’
general view of the world. In the relations among numbers they found
the essence of musical harmony, and they knew or discovered a number
of propositions that are mathematically significant. Is not this sedulous
preoccupation with number precisely what we mean by mathematics?
Do we not sce here, in epitome, the origin and basis of natural science?
To this question we must answer no. |
Number and mathematical science are by no means equivalent.
Numbers go back in origin to the mists of prehistoric times, but
mathematical science, properly speaking, did not emerge earlier than
sixth- and fifth-century Greece. People knew numbers before mathe-
matics in the strict sense; and it was in the pre-scientific era that the
“number mysticism” arose, or “number symbolism” or “numer-
ology,” which continues even now to exert a certain influence.2 No
one could overlook the fact that this kind of thing was present in
Pythagoreanism; Aristotle names first of all, among the ὁμοιώματα
which the Pythagoreans thought subsisted between numbers and
things, the equation of certain numbers with δικαιοσύνη, ψυχὴ καὶ νοῦς,
and καιρός (Met. 985b27ff), and only with a “furthermore” goes on to
add the mathematical theory of music. It is not necessary to attempt
here a complete presentation of Pythagorean number speculation; in
particular, the very luxuriant tradition from later antiquity can be left
* In English the hybrid “numerology” has become standard, though the term “arith-
mology,” introduced by Delatte, would be preferable on philological grounds. There is
no question of “mysticism” in the proper sense, involving union of the inner and outer
world; and the “symbolism” referred to is not an arbitrary assignment of “signs,”
but the discovery of apparently natural correspondences and interrelations (φύσει καὶ
οὐ νόμῳ, Philolaus fr. 9).—There is no comprehensive exposition of ancient number
symbolism, like that provided by Dornseiff for letter mysticism (Das Alphabet in Mystik
und Magie, 2nd ed., Leipzig, 1925). On particular topics, see E. Wélfflin, “Zur Zahlen-
symbolik,” ALL 9 (1896) 333-351; H. Usener, “Dreiheit,’” RhM 58 (1903) I-47,
161-208, 321-362 (cf. RAC s.vv. Drei, Dreieck). An exhaustive collection of material
is W. H. Roscher, Die enneadischen und hebdomadischen Fristen und Wochen der dltesten
Griechen (AbhLpz 21.4, 1903); Die Sieben- und Neunzahl im Kultus und Mythus der Griechen
(AbhLpz 24.1 1904); Die Hebdomadenlehren der griechischen Philosophen und Arzte (AbhLpz
24.6, 1906); Enneadische Studien (AbhLpz 26.1, 1907); Die Tesserakonten und Tesserakonten-
lehren der Griechen und anderer Volker (BerLpz 61 [1909] 17ff); Die Zahl 50 in Mythus,
Kultus, Epos und Taktik der Hellenen und anderer Vélker (AbhLpz 33.5, 1919); O.
Weinreich, Triskaidekadische Studien (Giessen, 1916) 95ff; Germain passim. V. F. Hopper,
Medieval Number Symbolism (New York, 1938, with a good summary of the ancient
phenomena).
466
FF “SF Serres Te
aside,® since the demonstrably ancient material, attested ly Aristotle,
is sufficient to establish the basic facts.
One is νοῦς and οὐσία δ two is δόξα: three is the number of the
whole—beginning, middle, and end; four is justice-—-cqual times
equal—but it is also, in the form of the tetractys, the “whole nature of
numbers”’;? five is marriage, as the first combination of odd and even,
male and female;® seven is opportunity (καιρός) and also Athena, as
the “virginal” prime number;® ten is the perfect number, which
3 For this tradition see, along with Zeller I 495-507, the works of Robbins and de Falco
(above, ch. I 3, n. 22). The Theologumena arithmeticae gives the most detailed summary,
and de Falco’s apparatus provides a convenient key to the parallel accounts. Posidonius
was involved in the transmission of Pythagorean number doctrine (Theo Sm. 103.16ff,
Sext. Emp. Math. 7.93ff; above, ch. I 3, nn. 8ff), but the crucial attestations are earlier.
Speusippus’ book Περὶ Πυθαγορικῶν ἀριθμῶν (fr. 4) was doubtless very important;
but there seems to have been an indeterminable number of developments and revisions,
many in pseudepigrapha (Prorus Περὶ τῆς ἑβδομάδος, Th. ar. $7.15, Megillus Περὶ
ἀριθμῶν, Th. ar. 34.21; cf. Jerome Epist. 49.19). The origin of Pythagorean mathematics
in number mysticism is emphasized by Joel, Geschichte der antiken Philosophie 1 (Tubingen,
1921) 3588, Rey 395; Schuhl, Essai 2588 Junge, DtMath 342f; van der Waerden, SA
93ff; Zeller (4958), but cf. 453 n. 3; he treats the symbolism as “development and
application” of the number theory, and Burnet refuses to see it as the starting point,
regarding the musical discovery as primary (EGP 107f).
4 The data given in Aristotle’s treatises are supplemented by Alexander from Aristotle’s
lost book on the Pythagoreans. At Met. 985b30 he speaks of ψυχὴ καὶ νοῦς, upon which
Alexander comments (39.13), νοῦν δὲ καὶ οὐσίαν ἔλεγον τὸ ev: τὴν γάρ ψυχὴν ws τὸν
νοῦν εἶπε (sc. Aristotle). That is to say, Alexander corrects Aristotle; by ψυχὴ καὶ νοῦς,
he means precisely νοῦς (cf. Pl. Crat. 400a: νοῦν καὶ ψυχήν, referring to Anaxagoras’
Νοῦς). This word was all he found attested; at the same time he supplemented Aristotle's
testimony with the word οὐσία; he is not paraphrasing, but commenting by the citation
of further source material—obviously Aristotle’s lost book on the Pythagoreans (ἐδήλωσε
[sc. Aristotle], Alex. Met. 38.8).
5 Arist. Met. 990223, Alex. Met. 39.16; τόλμα, Alex. Met. 74.12 (on Arist. Met. 990a22;
cf. Plut. De Is. et Os. 75, Th. ar. 9.6, Plot. 5.1.1, Procl. In Alc. I 104e). But here, Alexander’s
word φασίν apparently does not refer to Aristotle (75.27). For ἀνικία see Arist. Met.
g90a24, and cf. Th. ar. 34.11, 35.1.
8 Arist. Cael. 268a11. The Orphic verse, according to which Zeus is beginning, middle,
and end (Orph. frag. 21), is alluded to in the Derveni papyrus, and then in Plato (Leg.
71 56). The proclamation of the god is prior to the general Pythagorean formulation (cf.
also Pl. Parm. 1454), showing that in this respect Pythagoreanism is dependent on purely
religious, or “Orphic,” sources. Tannery (MSc If 184ff, HScH 390f) and Roscher (AbhLpz
1906, 18ff) supposed that there was an Orphic number symbolism earlier than the
Pythagorean; but the evidence cited is late (see Delatte, Litt. 208ff, on the supposed
Hymn to Number). Whether the use of number as an effective force went further in
Orphism than in other ritual, is quite uncertain.
7 Arist. Met. 985b29, 1078b23, EN 1132b23, MM 1182a1t, Alex. Met. 38.8ff; see above,
ch. I 3, n. 120, ch. II 4, nn. 154ff.
8 Arist. Met. 1078b23, Alex. Met. 39.8; cf. above, ch. I 2, n. 31. The interpretation of 6
as marriage (— 2 X 3) is later (Th. ar. 43.5ff, with the parallels cited).
® Arist. Met. 985b30, 990a23, 1078b22, Philolaus fr. 20 (above, ch. III 2, n. 52). According
to Alex. Met. 39.3ff, the number 7 is Athena, because among the first 10 numbers “it
neither begets another nor is begotten by any.” On the connection of gods and numbers,
see Xenocrates fr. 15, Plut. De Is. ef Os. 10, 75, Stob. 1 prooem. τὸ, Th. ar. passim.
467
VI. PYTHAGOREAN NUMARR ‘THEORY
comprehends the whole nutie of number and determines the struc-
ture of the cosmos, and with it ends the symbolic interpretation of
numbers!
Numbers are alternately even and odd, and this antithesis is the same
as that in the cosmos between “limiting” and “unlimited”: odd
numbers correspond to the more highly valued principle, that of
“limit,” and are also male, One is an exceptional case, being simulta-
neously even and odd, female and male. Geometrical figures, too, have
specific functions and their “character.” Philolaus assigns masculine
divinities to the triangle, and feminine to the square. It is probably
correct that the pentagram was a symbol of recognition among the
Pythagorcans, and the dodecahedron seems also to have played some
such role.¥
Of course there was nothing like a complete system; there were
overlappings and contradictions of all kinds—dvdyrn πολλὰ συμβαίνειν
τὰ αὐτά, as Aristotle says.!2 One might be tempted to ignore all this,
as a side growth on the main stem of Pythagorean wisdom; but more
careful examination reveals that this kind of lore is to be recognized
not as a branch but as a root, and one which goes very deep.
In primitive cultures, numbers are not abstract concepts with a
content which is mathematical and quantitative, but mysterious beings:
“each number has its own individual physiognomy, a kind of mystic
atmosphere, a ‘field of action’ peculiar to itself.’23 The comparative
material offered by anthropology need not be considered here except
” Arist. Met. 986a8, Probl. 910b31. The later compendia only treat the numbers up to
10, with the exception of Aristid. Quint. 3 p. 121ff M.
" Above, ch. I 2, nn. 24-32: the astrological derivation at Sext. Emp. Math. §.7f is
later.—Above, ch. IV 3, n. 60; II 4, n. 81; VI 3, n. ὅς,
' Arist. Met. tog3at. See also Zeller I 501.2. As an example, should we regard the
* perfect” number 10, which climaxes the serics, as even, and therefore unlimited?—
Phe ironic collection of groups of 7 at Arist. Met. 1093a13ff is regarded by Roscher
(AbhLpz 1906, 25ff) as ancient Pythagorean doctrine, which Aristotle took froma written
Pythagorean source. But the “Seven against Thebes”’ sounds like plain mockery, and the
“seven vowels” cannot be carly Pythagorean, at least in Magna Graccia (o foe w, 1G
XIV 636, 664, 647, from Metapontum; ε for 7, IG XIV 630, 631, 643, 652; H= h
ICs XIV 643; but H ἢν IC: XIV 648, 4th century B.C.). Roscher is 160 hasty in acceOtnG
av “doctrine” what may be no more than jesting.
MT. Lévy-Beuhl, Les fonctions meutales dans les sociétés inférieures (Paris, 19§1°), 236
avtr. by L. Clare, How Natives Think (London, 1926) 206. The section 235-257 = 225-
221 ts entitled “ Puissance mystique du nombre” (“* Primitives’ Numeration”), See also
Ebert, Reallex. ας Vorgesch. XIV 475ff: Ruble in Kittel, Theol. Worterb, des N'T (and Eng
tr), 8.v. ἀριθμός, with references: A. B. Keith in Hastings, FRE IX (1917) 407ff: A.
Seidenberg, “The Ritual Origin of Counting,” Archive for Hist. of Exact Sciences 2 (1962)
1-40,
468
4. Number and Cosmos
insofar as we find phenomena similar to the Pythagorean, Notthat we
start by assuming connections and influences, but that the peculiarities
of certain psychological phenomena, more likely to be obscured by
concepts of Greek science and mathematics, may be clarified in this
manner.
In certain Negro peoples the king of the gods is invoked thrice; the
queen four times; the amulet worn by a man has 3 knots, that of a
woman 4. The birth of a boy is celebrated after 3 days, that of a girl
after 4; a dead son is mourned for 33 days, a daughter 44. If we add that
the number ¢ plays a part in the marriage ritual, we have 3 as the male
number, 4 as the female, and 5 as the number of marriage—among the
Negroes of the Sudan!!4 This sounds so Pythagorean that one is almost
tempted to think of some kind of direct connection. In fact, this
possibility cannot be excluded entirely; but the same situation—3
as the man’s number, 4 as the woman’s—is found in the ritual of
Alaskan Eskimos.!® At other points on the globe’s surface, 4 belongs
to the man and 3 to the woman; and the numbers 9 and 7 are found
for man and woman, as well as 3 and 5, 4 and 5, and 6 and 5. The range
of the evidence extends from the South Seas, over Africa, to the
American Indians. There are traces of a similar use of 3, 4, and 5 among
the Germans;!’ and in Rome a girl was named 8 days, a boy 9 days
after birth,1® while 5 torches were used in the wedding ceremony.!®
Above all, we find that with very few exceptions festivals of the gods
were set on odd days of the month.?°
14 Ἐς Fettweis, “Ueber das Verhaltnis des mathematischen Denkens zum mystischen
Denken auf niederen Kulturstufen,” Archeion 14 (1932) 207-220; “‘ Berithrungspunkte
der pythagoriischen [sic] Zahlenlehre mit dem Totemismus,” Zeitschr. f. philos. Forsch.
5 (19$0-1951) 179-196.—The Arabs of contemporary Egypt believe that a grave with
three cover slabs is that of a man, one with four that of a woman (Fettweis, p. 188).
15 Fettweis, Zeitschr. f. philos. Forsch. 1950-1951, 179; Archeion 1932, 210f, referring
to Plut. De Is. et Os. 56 (3 is Osiris, 4 Isis, 5 Horus). Plutarch combines Egyptian and Greek
material; there was certainly some Hellenistic Greek influence in the Sudan.
16 Fettweis, Zeitschr. f. philos. Forsch. 1950-1951, 190.
1 H. von der Au, “ Drei larren Strémp: Zur Deutung eines Vogelsberger Frauentanzes,”’
Hess. Bl. f. Volkskunde 35 (1936) 73ff, on the number § and fertility magic; p. 83 on the
number 3. See also J. Grimm, Deutsche Rechtsaltertiimer 14 (1899) 285ff.
18 Plut. Quaest. Rom. 102, Macrob. Sat. 1.16.36, Festus p. 120 M. Plutarch naturally
interprets this as Pythagorean; in the Ist century B.c., Castor of Rhodes connected
Roman and Pythagorean (FGrHist 250F16).
19 Plut. Quaest. Rom. 2. The use of the word cerio (κηρίων) shows that Greek influence
was present.
20 Exceptions: the Regifugium, the Equirria, and the procession to the Argei (see G.
Wissowa, Religion und Kultus der Romer [Munich, 19127] 436f, Censorinus 20.4, Festus
p. 109 M.). T. Mommisen sees Pythagorean influence (Die rémische Chronologie [Berlin,
1859*] 15.12), as does Ε, BOmer, Ovid, Fasten I (Heidelberg, 1957) 35. ΝΜ. Aly wishes to
use this to date the establishment of the calendar of festivals (ARW 43 [1936] 59); and
469
. PYTHAGOREAN NUMBRR THEORY
When Babylonian seers name the 13th, osth, and ith days as
favorable for setting a ridgepole™ they are differentiating the odd
numbers, as favorable, from the even. In East Prussia odd days of the
month used to be regarded as lucky, in Pomerania the even.22 In
ancient folklore, a remedy for headache was made by crushing an odd
number of berries ;#" it was recommended that a hen be set to hatch an
odd number of egys,#4 and that a herd of animals consist of an odd
number.” Here Pythagorean influence cannot be ruled out,2* but the
basic idea is older, and the practices are independent of Pythagoras.
The number 7 has a special significance in ancient Egyptian medi-
cine,?? as in that of certain Indian tribes.28 Solon uses it in dividing
human life into periods,?® as do the Etruscans.2° Among the Babylo-
nians, each divinity has a specific number which belongs to it—for
example, Ishtar has 15, perhaps reminiscent of the ς which is γάμος .51
ἐπ -- τὸ τ τς gs ὃ 0
ancient tradition, relying on just this kind of observations, made Pythagoras the teacher
of Numa (on this, see Burkert, Philologus 1961). See also above, ch. II 4, nn. 128, 136,
On the other hand, Macrobius (Sat. 1.13.5) speaks of “‘secretum hoc et ante Pythagoram
parturiente natura.’’ K. Latte, Romische Religionsgeschichte (Munich, 1960) 199, called
attention to the fact that with the reverse numeration of days, used by the Romans, these
festivals come on even-numbered days in the latter half of the month; but it may be
that the reverse numeration was not the oldest (A. K. Michels, The Calendar of the Roman
Republic [Princeton, 1967] 139).—The role of 3, 9, and 27 in the Roman cult of the dead
is contrary to Pythagorean doctrine (H. Diels, Sibyllinische Blatter [Berlin, 1890] 408);
cf. below, n. $6. Thus in any case the roots of Roman number symbolism are earlier than
Pythagoras.
Ἢ Meissner II 2788.
δὲ Handworterbuch des deutschen Aberglaubens VII τό4τῇ.
* Plin. HN 23.156; cf. 24.82, 28.56, 28.33 (“Pythagoras’’),
4 Varro Rust. 3.9.12, Columella 8.5.8, Plin. HN 10.1§1, 163, 18.231, Pallad. 1.27.1,
Creop, 14.7.13. See Roscher, AbhLpz 1904, ὅς n. 153. This is common in Germany:
Handwarterb. des deutschen Abergl. IV 452.
Ὁ Geop. 18.2.8; also attested for Germany, Relig. in Gesch. u. Gegenwart V? 2068.
“On Hellenistic pscudepigrapha dealing with georgica and related topics, see M.
Wellmann, “Die Georgika des Demokritos,” AbhBln 1921 n. 4: “Die φυσικά des Bolos
Demokritos und der Magicr Anaxilaos aus Larissa,” AbhBln 1928 n. 7. Their influence
is scen as carly as Cato Agr., Wellmann 1921, 34ff.
“7 Papyrus Ebers, ca. 1500 B.c.; Roscher, AbhLpz 1906, 108.
441. Lévy-Bruhl, How Natives Think (above, τ. 13) 212.
"fr. 1g Diehl; often cited in ancient nucrology, where the treatment of 7 was
generally the fullest.
™ Varro ap. Cens. 14.6, Varro, who likes to “Pythagorize,” is of course not entirely
tcliable on such a matter. He had himself buried “ Pythagorico modo” (Plin. HN 35.160).
"Meissner 131; cf. also A. Jeremias, Handbuch der altorientalischen Geisteskultur (Berlin
ty2y") 2041. FX. Kugler, “Der Ursprung der Zahlsymbole . . . in pythagoreischer
Beleuchtung,” ΚΙ αἰ (1911) 481-496, wished to use this as an indication that the whole
of Pythagoras’ number symbolism was derived from Babylon; but the development is
a complex one, and number symbolism can be shown to have existed in Greek culture
before Pythagoras (Germain 61ff).—The highest god of the Hittites, the weather god,
is represented ideographically by the sign for τὸ (W'orterbuch der Mythologie ed. H.W.
Haussig |Scuttgart, 1961] 1 209). |
470
4. Number and Cosmos
The number 4 plays a dominant role in the cosmology of the American
Indians: corresponding to the 4 directions and the 4 winds, the pods
are divided into groups of four, and colors, actions, and tribal terri-
tories are divided according to the same principle.®# It is number that
orders the universe.
In Chinese thought, number symbolism was developed into an
amazingly delicate and complex system, which comprchends cosmos
and man, nature and social order.** The basic antithesis of Yang and
Yin is manifested in the antithesis of odd and even number; and it is
odd number which corresponds to the active, male principle, Yang.
The significance of the numbers from 1 to 10 is of fundamental
importance. They are divided into two groups of five, from 1 to § and
from ς to 9, and then brought into connection with the five elements,
the points of the compass, the seasons, the main types of taste or flavor,
the colors, and, above all, with the five notes of the pentatonic scale.
Five is the number of procreation. The numbers are grouped together
in a figure made with counters or pebbles—i7do0.—which is a represen-
tation of the entire universe.?4 The Chinese know the numerical laws
of the musical intervals, and express the relative height of a tone in
terms of a series of numbers, which is also used in determining lengths
for bamboo flutes. But this is not a matter of physical theory, but of
analogies with cosmic regularities; in the ratio 3 : 2 or 4: 3 is expressed
the relationship of Yang and Yin;® the numbers that occur have a value
and significance of their own. Unity, in this context, is not regarded as
a number and can be added or subtracted at will,3* in order to produce
a result in which the relationship in question can be expressed in
‘significant’? numbers. Thus, for example, the interval of a fourth can
be 81: 60 instead of 80: 60; and the arithmetical complications in the
calculation of the scale, in which the Pythagoreans got themselves
tangled, can readily be avoided. The objective is not, in fact, accuracy,
but the revelation of connections and correspondences. The five notes
of the pentatonic scale are also arranged in the form of a pentagram.*”
82 Lévy-Bruhl, How Natives Think, 210; Schuhl, Essai 259 ἢ. 5.
88 For the following see Granet, esp. pp. 209ff; H. Koster, Symbolik des chinesischen
Universismus (Stuttgart, 1958) 44ff; Fung Yu-Lan, A History of Chinese Philosophy, tr.
D. Bodde (Princeton, 1952-1953) II 93ff. For references and advice in this field, I am
much indebted to Prof. H. Steininger.
84 Arrangement Ho-t’u, Granet 177f; connected with the Ming t’ang, Granet 178ff.
35 See above, ch. V I, n. 64, on superparticular proportions.
36 Compare the German “acht Tage” for a week, along with
““quinze jours”) for “2 weeks” (and English ‘ fortnight”).
3? Granet 230ff.
‘14 Tage” (French
471
eee 8 8 δ Αἰ ΥΟΦΙΒΗΚ ΤΗΒΟΝΥ͂
The Chinese knew the " Pythapotean theorem,” but used it only with
whole numbers, Here again the unit does not count, so that 8, 9, 12
can represent the sidew of a right triangle,?* and 5, 5,7 those of a right
isosceles triangle.” ‘Ihe objective, again, is not mathematics, but the
aim of making the proportions in the roof of a Chinese house corre-
spond to the proportions of the cosmos,
We read in a text of the Han period which features the study of
“emblems and numbers,” “| lcaven, earth, the yinand yang, and wood,
fire, earth, metal and water, make nine; together with man, they
make ten. Heaven’s number is with this made complete.’’4° It is as
though we had an ancient Pythagorcan text before our eyes. Also, as
the Platonic “system of derivation” follows the Pythagorean, in
China a philosophy emerged, neo-Confucianism, which is strongly
reminiscent of Platonism,*
Perhaps one ought not to exclude completely the possibility of
direct connections between the Pythagoreans and the Chinese:42 but
whatever tenuous lines of connection between East and West there
were before the beginning of the modern age, they are not so important
as the basic, underlying idea or attitude. Speculation about numerical
relationships in the cosmos is world wide, firmly established in ritual,
and capable of being elaborated into a rich and igeniously structured
system without either presupposing or giving rise to mathematics in
the proper sense of the word. To suppose, therefore, that number
speculation is derived from mathematics and that consequently Pythag-
orean mathematics is primary (as others have done besides Aristotle)
is no better than a petitio principii.
The nature and origin of number symbolism has also been treated
98 82 + οἱ — 122+4+ 3: Granet 240
88 Plato calls 7 the διάμετρος ῥητὸς πεμπάδος (Rep. 546c); cf. the Pythagorean “‘side
nuinbers”’ and diagonal numbers,” above, ch. VI 2, Ἡ. 14.
“ Fung Yu-Lan II 19 (Tung Chung-shu, 2nd century B.C.); cf. the text Ta Tai Lichi,
cited by Koster 45f; “Heaven one, earth two, man three.”
“' Fung Yu-Lan II 458 cites Shao Yung, 12th century a.p.: “The Supreme Ultimate is
a unity which does not move. It produces a duality, and this duality is spirit... Spirit
produces numbers, the numbers produce emblems, and the emblems produce imple-
ments”’ (i.e., the objects of the physical universe). This is the series ἕν, (ἀόριστος) Suds,
ἀριθμοί, εἴδη, αἰσθητά.
42 Granet is negative (209ff), but Fung Yu-Lan (II 11) holds such an influence to be
possible. Earlier, Pére Amiot believed that Pythagoras was dependent on the Chinese
(Mémoires concernant Vhistoire, les sciences . . . des Chinois [Paris, 1776-1791] VI 173 [not
accessible to me]). E. Chavannes reversed the relationship (Les mémoires historiques de
Se-ma-Ts’ien ΠῚ 2 [Paris, 1895] 630-645, cf. 647 on the “Metonic” cycle). Hellenistic
astrology reached as far as Japan.
472
4. Number and Cosmos
from the point of view of C, G. Jung’s school of paychoatial yain The
mass of data collected is amazing; number syinbolinn continues to
exert a surprising influence even in the subconscious mind of modern
man. The fascination of the “perfect” number 3 was exploited by the
Third Reich; and the pentagram lives on in the flags and camblens of
the United States and the Sovict Union.“ Thus for the payclruolopint
the numbers are archetypes. To be sure, the connection of mind and
symbol does not yet seem to be completely clear or capable of formula-
tion with scientific precision. The meanings of the individual numbers
are sometimes almost uniform, but sometimes entircly ambiguous,
giving the impression of being arbitrarily interchangeable. It is surely
mistaken to derive number symbolism from certain particular natural
observations, as Roscher referred the special significance of 7 and 9 to
the phases of the moon.** It is also clear that certain elementary calcula-
tions play an important role. One learns to count and calculate in
childhood, and from the beginning the numbers are apprehended as
things, with certain characteristics-46 they preserve this peculiarity
even, at an unconscious level, in the mind of the adult. For example,
the odd numbers, which cannot be divided evenly, are felt as uncom-
fortable, even perverse—and therefore, if one knows how to use them,
powerful and lucky. A prime number like 7 is especially hard to
manage and therefore very significant; and here an additional psycho-
logical factor is at work—it is easy to picture in one’s mind a group of
six, or a hexagon, but much harder to imagine a heptagon. Thus many
kinds of threads join to compose this complicated fabric.
Like the numbers, the simpler geometrical shapes have an “arche-
typal” symbolism. The square symbolizes the carth both for the
Chinese and for certain Indian tribes;4” and Philolaus “dedicates” the
angle of the square to the goddesses Rhea, Demeter, and Hestia.*# But
here too the attributions are inconsistent; there is no direct relationship
“L. Paneth, Zahlensymbolik im Unbewusstsein (Ziirich, 1952); see also C. G. Jung,
Aion (Zurich, 1951) 323f.
44 Paneth, 45ff, discusses the significance of ς as the number of vitality and sexuality,
without knowing the Pythagorean equation of ς and marriage. For the pentagram on
flags, see De Vogel, Pythagoras, 297-299.
45 Roscher, AbhLpz 1904, 67ff. On the other hand, H. Quiring tries to trace the
importance of 7 to the planets alone (Altertum 4 [1958] 208-214); correctly Junge, DtMath
6.
a Dornsciff (above, n. 2), 17ff, uses the expression Kindheitsmystik.
ὁ Lévy-Bruhl, How Natives ‘Think (New York, 1966) 114 (Dakota Indians), They also
represent the earth by four dots (::).
* Above, ch. ΓΝ 3, αν 60, Eudoxus adds Hera and Aphrodite.
473
eee Φ ὁ ΦΑΣΕΧ ΦΥ͂ ἐΥ̓ΠΗΝ LT TIN ΚΥ
between the triangle asa female sexual symbol and the male gods to
whom the Pythagoreans dedicated the angle of the triangle (above
n. 48). It is noteworthy that in prehistoric ornamentation there are
examples which anticipate the “tetractys.”®" The ψῆφος figures, too,
with their speculative interpretation, “make sense,” ina way, as
“archetypal” patterns; they “speak to”? certain psychic dispositions.
Those mysterious dodccahedra mentioned above® may be rooted in
similar soil.
Greece, too, has its primeval, ritually significant symbolic numbers.
Even in the Homeric epics one notices the preference for certain
numbers, ®2 especially 3 and 9, and also 5 and 7—the odd numbers.
The number 8 hardly occurs at all, and also apparently played no part
in the ancient Pythagorean tradition.53 Certain numbers belong to
certain gods; the cult of Apollo and that of Dionysus were dominated
by the numbers 7 and 9.54 The significance of 3 in purification ritual
was emphasized by Aristotle; as Usener perceived, it is rooted in a
primitive conception of number, in which, after 1 and 2, 3 means
plurality in general.5° At the symposium one set out three craters of
wine, and poured a libation from the first to Zeus and the Olympians,
from the second to the Heroes, and from the third to Zeus Soter or
Teleios.5* Two belongs to the chthonic world, whereas 3 is the number
4 E.g., the expression δέλτα in Ar. Lys. 151, Paus, 2.21.1; A. Stuiber, RAC s.v. Dreieck,
with refs.; M. Eliade, The Forge and the Crucible, tr. S. Corrin (London: Rider, 1962) 185.
°° A. Ὁ. Nock, AJA §0 (1946) 142 n. 14; M. Hoernes and O. Menghin, Uregesch. d. bild.
Kunst in Europa (Vienna, 1925%) 49 (rattle amulets from Tribano, near Padua), 283, 349,
397, 429, $59. See also Stapleton, Osiris 1958, 32ff: J. Schwabe, Antaios 8 (1967) 444
(pottery from Samarra, East and West 1 4 [1964-1965] 72); J. Meysing, Rev. des sciences
religieuses 39 (1965) 218.
51 Above, ch. VI 3, n. ὅς.
52 See Germain, passim.
58 Cf. nn. 4-10. For Homer, Germain 8f ; on the other hand, 8 becomes important in
late antiquity and in Christianity: F. J. Délger, Antike und Christentum 4 (1934) 153-182.
δ Roscher, AbhLpz 1904, 5ff, 54ff For example, the Carneia lasted 9 days; 9 men
gather in each of 9 “sunshades” (tent-like structures: Ath. 4.141e); Apollo was born on
the seventh day of the month, and he is ἑβδομαγέτας at Aesch. Sept. 800. There is a 7-day,
sevenfold beating with κράμβη ἑπτάφυλλος at the Pharmakos ritual (Hipponax fr. 11;
IX 49f Diehl-Beutler; cf. Nilsson I 107ff). There are 7 bunches of grapes and 7 dolphins
on the famous Dionysus cup of Exekias.
ὅδ Above, ἢ. 2; Arist. Cael. 268a14; cf. Od. 11.28. Reemphasizing 3, one gets 9 and 27,
which are important in the cult of the dead (Soph. OC 470ff, Germain 38ff; on Roman
phenomena, above, n. 20).
56K. Kircher, Die sakrale Bedeutung des Weins im Altertum (Giessen, 1910) 17ff, 34ff:
Nilsson, Op. I 428ff. Ζεὺς τέλειος, Eur. fr. 148, Ar. fr. $26, Philochorus FGrHist 328F87
(with Pythagorean interpretation). In Pythagorean doctrine, one is enjoined to bring odd-
numbered offerings to the gods of heaven, even to those of the underworld (Plut. Numa
474
4. Number and Cosmos
of completion or fulfillment. If one is tempted to see here the early
onset of Pythagorean Interpretation, still even in Homer linaselt, dee)
means “doubt,” or confusion’: and it is not far from here to δύξα, In
wrestling, a match has been ended, since time Inunemorial, when one
contestant has won three falls (τριάζειν). The special status of the first
four numbers, as a group—the tetractys—is reflected even in the fret
that they alone are declined, after Indo-European tradition,
According to Aristotle,®’ the Athenians were divided, in carly times,
into 4 phylae and 12 phratries each made up of 30 families, on the made!
of the division of the year into seasons, months, and days. Numerical
order unites society and cosmos; and even if this were to be put down
as fourth-century speculation, it may well be based on something older.
The number 7 is in Solon’s view the one according to which human
life is to be divided into periods; and neither its interpretation as καιρός
nor its important role in Greek medicine can be traced to Pythagoras
as originator.°* The most popular expression of the inferior status of
even numbers came in the theory that 7-month and 9-month babies
can live, but 8-month babies cannot.®9 It is asserted in the Hippocratic
book on the subject that the first movement of the embryo in the
womb takes place after 3 months in the case of boys and 4 in the case
of girls.6° The formation of the embryo takes 30 days for boys and 42
for girls; and the postpartum discharge lasts a proportionate length of
time. Thus 3 is the male and 4 the female number, which in the
second example is conjoined with a period measured in sevens. There
is some Pythagorean material in the Hippocratic writings;® but the
eee
14, Por. VP 38, lam. VP 156, Schol. A Ji. 23.171; Plato alludes to this, Leg. 717a; Serv.
Aen. 3.305, Buc. 5.66, 8.75, ascribes it—probably following Varro—to the Romans),
But this is not universally valid for either Greek ritual (above, n. 55) or Roman (above,
n. 20).
5? Ath. Pol. fr. ς = Lexicon Patm. p. 1§2 Sakkelariou.
58 Above, ἢ. 29. For the material in the Hippocratic corpus about ‘‘critical days,” see
Roscher, AbhLpz 1906, 55-84. Hippocrates (Epid. 1.26) distinguishes a number of κρίσιμοι
ἐν ἀρτίῃσιν from another ἐν περισσῇσιν. See also Heraclitus A18, B126a (whose genuine-
ness is very doubtful, because of the dual, quite irregular for Ionic), Alemacon ΑἹς,
Empedocles A83, fr. 1532, Hippo A16, Aristox. fr. 23. Criticism of Pythagoricé numeri:
Cels. Med. 3.4.
5° Modern medicine does not confirm this doctrine, at least in such an extreme form.
Of course, ancient physicans were very seldom able to determine precisely the time of
conception.
°° Hippoc. Nat. puer. 21, VII 510 L., Steril. 233, VIII 446 L.; differently Diogenes ον
Apollonia A26, Bg.
6! Nat. puer. 18, VII 500 L.: but according to Diocles and Strato the embryo is formed
in the case of girls by the sixth, in that of boys by the seventh hebdomad (Strato, ἔτ,
97-98 Wehrli)—here the girls are quicker, but they still have an even number,
* Above, ch. II 2, nn. 113 fF; ΠῚ 3N. 63, 86.
A7S
1, PYTHAGOREAN NUMBER THEORY
forms ὁ τρίπους and ἡ τράπεζα are attested even in Mycenacan Greek,
and there is no obvious reason for the variation in gender in nouns
from the same root, except that pender is determined by number. The
birthday of Apollo is celebrated on the seventh, that of Artemis on the
sixth day of the month. A man, says [esiod,® should marry at about
39,
ἡ δὲ γυνὴ τέτορ᾽ ἡβώοι, πέμπτῳ δὲ γαμοῖτο.
A girl has four years to herself after she reaches puberty, and her
marriage is to take place in the fifth. So the numbers 3, 4 and 5, as
those of man, woman, and marriage, are present in early Greek folk
custom.
All this goes to show that the curious numerical correspondences,
known as Pythagorean as early as Aristotle, are not a late development
or a trivialization of a Pythagorean philosophy or mathematics, but
have their roots in primitive ideas of number. The numerical symbols
for male, female, opinion, marriage, opportunity, and “the whole’
are carlicr in origin than the time of Pythagoras; they were present not
only in oricntal lore but in the language and customs of the Greeks
themselves. The notion that numbers have a “metamathematical,”
cosinic significance, and that they reveal the principle of the order of
the world and of human life, is not any kind of scientific or philo-
sophical insight, but a readily comprehensible characteristic of pre-
mathematical thinking about number. Pythagorean number symbolism
is therefore much older than any natural science, mathematics, or
astronomy that Pythagoras or his pupils could be imagined to have
practiced. It has nothing to do with science in our sense—which is to
say, the Greek sense—of the word; it neither presupposes this nor
691)... 2.44; Deubner, Aftische Feste (Berlin, 1932) 179, 201, 209.
44 Op, ὁυς, 6y8. Naturally, 40 cannot be the usual age for women to marry, nor, hardly,
can 17 for mens it is all the more remarkable how 3, 4, and 5 are worked into the formula-
tion. A provision in Plato’s Laws is that a man may be punished with blows until the
age of 30, a woman till 40 (932b-c, cf. 845c). According to the Republic (460e), a woman
may have children from the twenticth to the fortieth year, a man from the twenty-fifth
to the fifty-fifth.—Perhaps it is worth noting that though groups of 4 are uncommon in
ritual, it is in women’s cults that they do appear (Germain 51; the Heraea at Olympia,
Paus, 6.16; Demeter Chthonia at Hermione, Paus. 2.35.7; the four daughters of Celeus,
Hynn. Hom, Cer, 1ogf).— One hopes that, on the third day of the month, a child born
will turn out to be a boy; fora girl born on this day would turn out to be unfeminine in
character (Schol. BT on If 8.39, Suda s.v. Τριτογένεια). Naturally, there are exceptions:
Heracles and Hermes were born on the fourth.—One resists the temptation to mention
the 3 Doric and 4 Ionian phylac.
476
4. Number and Cosmos
advances it, The Pythagorean doctrine that “all is number prows
directly out of “archetypal” number symbolism, which tn one depree
or another is worldwide in occurrence.
Of course, number symbolism can be combined with scientitte
knowledge. Both in China and among the Pythagoreans, it took over
mathematical music theory; and in the “harmony of the spheres" we
have an adaptation of mathematical musicology to the newly dis-
covered, scientific understanding of the cosmic order. But this cannot
mean more than an application and confirmation of a basic idea which
was already present and was open to any kind of enrichment. Number
is not quantity and measurability, but order and correspondence, the
articulation of life in rhythmical pattern, and the perspicuous depiction
of the whole as the sum of its parts. To see a “consistently quantitative
view of the world’’® in Pythagorean number theory is a mistake.
One cannot help recognizing how closely this number symbolism is
connected with the realm of the acusmata. In both cases the origin lies
in primeval custom, found in somewhat similar form among the most
varied peoples and cultures; and in both cases we find the circumscrip-
tion of life and thought by forms and formulas, which are laid on the
Pythagorean by authority and whose explanation and interpretation
are secondary in importance. Number in fact means restraint; counting
is performed in successive acts, and thus time itself, composed of
successive events, is number.** The great mass of the acusmata had to
do with sacrificial ritual, and its methods and times, the καιροί, 57 so
that correct piety (εὐσέβεια) depends on knowing number. While this
reveals the orderly arrangement of time, such a point of view may lead
also to thoughts of the recurrence of the same, of periodic transmigra-
tion, or even to the recurrence of the same condition of the world.®®
To the four seasons correspond the four ages of man’s life;®® the
65 Frank 72: “konsequent quantitative Weltanschauung.”
66 It is not a coincidence that in the oldest references τετράς, δεκάς, τριακάς are mostly
used in a temporal sense, of the 4th, roth, and 3oth day (Hes. Op. 794, 798, Hymn. Hom.
Merc. 19; differently in Il. 2.126). Seidenberg (above, ἢ. 13) propounds a thesis that
counting had its origin in a “creation ritual.” This is unprovable, but his collection of
material on ritual counting is worth attention.
67 Jam. VP ὃς (following Aristotle: above, ch. II 4, ἢ. 5): τὰ δὲ πλεῖστον ἔχοντα μῆκος,
περί τε θυσίας καθ᾽ ἑκάστους τοὺς καιροὺς πῶς χρὴ ποιεῖσθαι. A portion of the exposition,
based on later sources, is at Iam. VP 152.
68 Eudemus fr. 88, Dicaearchus ap. Por. VP 19.
69 This is frequent in later sources: the “‘tripartitum,” D.L. 8.10, Ov. Met. 1§.199-213.
For parallels see Delatte, Vie 110. Perhaps Alcmaeon has this in mind in fr. 2, if it means
that, while the year is continually renewed, the winter of man’s life is not followed by a
new spring (above, ch. III 3, n. 97). Cf. the riddle of the Sphinx solved by Oedipus, and
Hippoc. Nat. hom. 2.
477
ἊΨ =F = = 2 κ'ῷ ee 2 ei eee OS FT oY wraive δ δὲ ὃ i IORI
cotrespondence of part and whole, of man and cosmos, is mirrored in
number, To the order of time corresponds also the order of space.70
There are some indications that the ancient idea of macrocosm and
microcosm was known to the Pythagorcans;7! we “enumerate”??2
the parts in cither realin, and it is number that reveals their correspond-
ence. Thus the multiplicity of the world is reduced to clearly articu-
lated groups, whose mutual relationship is known to the “wise
man’; and he “honors” this orderly arrangement in his practical
activity.
It is not clear to what extent the idea of number was dominant in
the acusmata. Only two of the individual precepts we have speak of
number: the catechetical “ What is wisest? Number” and the reference
to the tetractys, In this word there seem to be condensed a great many
connotations. Four has its own value and significance as the number
of “right” ordination and adjustment (above, n. 32); the triangular
figure, made up of ψῆφοι, speaks for itselfas.a graphic and diagrammatic
representation of order. That the first four numbers add up to ten, a
number that had drawn special attention from very early times, is
immediately evident from the figure; and it may well have seemed the
supreme manifestation of the mysterious power of the tetractys that
the same first four numbers express the basic ratios of the musical
intervals.
In any case, there is no break in level between the number symbolism,
even in its cosmic sense, and the other acusmata. Number, ritual, and
the doctrines about the soul, motifs which compete in the later course
of intellectual history, coincide in Pythagoreanism—in a completely
pre-scientific realm. If this content of the Pythagorean tradition is
unmistakably older than Pythagoras, then it is quite improbable, and
would have to be proven beyond doubt, that all of this worked its way
into Pythagoreanism by some sort of roundabout route, rather than
coming by way of the master’s own teaching. Our conclusion must be
that in number symbolism, as in the acusmata, we are dealing with
doctrine of Pythagoras himself.
The question remains, wherein lay the incontestable fame of the
man and his influence on generations of disciples; must he not have
contributed something basically new, for example by the “recognition
© Hesiod measures cosmic distance in time (Th. 722£), Anaximander in linear terms
(above, ch. IV 1, nn. $5-56).
"1 Above, ch. I 2, nn. 45ff; III 2, mn. 1§0, 162.
12 καταριθμοῦμεν, Diod. 1.11 (Hecataeus of Abdera?), describing the “Egyptian”
doctrine of microcosm and macrocosm.
478
be ΝΉΝ and Coonos
of the universal regularity, to be comprehended by meana of number,”
through which the word cosmos acquired “a deeper meaning, and
one of decisive importance for the history of philosophy"? Mathe-
matical astronomy, after some stirrings in the fifth century, was only
brought to full development by Eudoxus; music theory wan at first
more a number game than a science; and the “philosophiae naturalis
principia mathematica,” in the sense of Newton, were never attainable
to the Greeks, even though in the Timaeus it seems that Plato dreamed
of something of this kind.” Modern perspectives distort our view of
the ancient “wisdom” of Pythagoras, as in fact it had soon become
distorted in antiquity. An impressive achievement can consist in the
synthesis, systematization, and consistent development of something
preexisting, to say nothing of the unaccountable workings of a
personality which is not dependent on originality or the independent
importance of specific thoughts or doctrines, but of itself enhances the
significance of meditation and teaching. Pythagoras was no lonely,
unrecognized thinker, but an extremely renowned and successful
“sage,” the founder of a society which lasted a long time. Further, is it
not a great achievement if a strong personality, by virtue of its charis-
matic power, gives new life to what is old, and preserves its spiritual
forces through a period of turmoil and change? The schemes and
projects of science always fall short; there remains an unexplained
residue to cause disquiet and the yearning for completeness. To round
this out into a comprehensive knowledge which could satisfy all the
aspects of a man’s being, this ideal was regarded as accomplished by
Pythagoras. Perhaps it is here that we should see his great importance;
because of him there remained alive something of the ancient lore,
pushed to one side by the growth of science but never really dispens-
able, powers of the past which could preserve a feeling of security in
an increasingly secular and material world.
It has long been known that conscious and unconscious, rational and
irrational impulses, logic and mysticism, interpenetrate in a compli-
cated and nearly inextricable fashion. As Kepler discovered his second
planetary law in “‘Pythagorean” manipulation of regular polyhedra,
so one might find it obvious that precisely the pre-philosophical lore
of Pythagoras provided the stimulus for Pythagorean science. But not
only does the cosmic significance of number come from pre-logical
"8 Kranz, ABG 1955, 32: “eine tiefere, und zwar fiir dic Philosophiegeschichte ent-
scheidend wichtige Bedeutung.”
"4 See Becker, Fs. Gadamer 12.
479
VI. PYTHAGOREAN ΝΌΜΒΠΕ THEORY
number symbolism, but, even in that which Aristotle presents as the
philosophy of the Pythayoreans, there cmerges again and again a spirit
and method directly opposite to that of exact mathematics, so that the
latter cannot have arisen from the activities of the Pythagorcans. It is
not an unbroken unity of science and religious-cthical teaching that
we find in the Pythagorean tradition, but a groping attempt to mediate
between two levels, to transpose an ancient interpretation of the world
into the language of the recently founded φυσιολογία. In this transposi-
tion, apparently, the opposition of “limiting” and “unlimited” first
attained an important part, even though the high valuation placed on
“limit” is a derivative from the ancicnt lore. This made it possible to
show a relationship between essential characteristics of being and the
world of number. Other agreements were sought and happily found—
significant numbers in the calculation of the scale, regularities in the
pebble diagrams, orderly behavior among the stars. But there is no
steady continuity or consistency; different kinds of material were
present, partly retained, like the ancient numerical symbols, or the
mere juxtaposition of analogous ideas,’> and partly added, like the
medical doctrines in Philolaus (A27-28), which do not have any
necessary relation to his cosmological ideas. The acusmata and similar
regulations were passed on, without further justification, as Plato attests
Philolaus did (Phd. 61d-e). Thus this Pythagorean philosophy was a
synthesis closcly tied to the conditions of the time, and without any
lasting validity. The nature of the tradition authorizes us to regard it as
largely the work of Philolaus, following after some similar attempts of
Hippasus.
But even Philolaus and his pupils must have been persuaded that in
their thoughts they were only following and carrying out the insights
of Pythagoras, and presenting in different terms what he had long ago
known and meant. New interpretation of this sort is inevitable, if an
authority is to maintain his position as intellectual history progresses.
Pythagoras was the wisest of all men, the tradition said. For an age that
still knew no science, γόητες and τελεσταί were the “sages,” and
Pythagoras belongs in this context, as we can see from the oldest
witnesses, whether they mock or admire him. But only a few decades
later, it was impossible to accept this; so the image of Pythagoras must
change. If Pythagoras was “wise,” he was “wise” in the spirit of the
new age, certainly not a γόης but a scientist. The “sage” acquires his
15 Philolaus fr. 13.
480
4. Nomber and Cosmos
knowledpe by his relation to the gods, whereas the aclentiat bases his
on deductive proof. ‘The “sage”? works immediately, in his own
person, whereas the scientist writes books. The "sage" knows the
roads through Heaven and Underworld, whereas the scientiat measures
cosinic distances in stades. The “sage” interprets the signs of the
zodiac, whereas the scientist calculates the movements of the heavenly
bodies in advance. Number lore in particular must inevitably appear in
a very different light; what had been a symbolic ordering and classifi-
cation of the multiplicity of phenomena becomes, in hindsight,
mathematics, for a nonmathematical idea of number is no longer
acceptable. The delight of the pragmatic historian in a πρῶτος εὑρέτης
helped to transmute this interpretation into a historical datum.
Plato forced the divergent factors of rationality and morality, of
physical science and speculative interpretation, into a new synthesis,”®
though he is quite consciously continuing ancient tradition. All the
more, it was Plato’s pupils who placed themselves under the authority
of “the ancients,” and as a consequence saw Pythagoras in the light of
what Plato himself had been the first to do. The pre-Socratic inter-
pretation of Philolaus, inadequate in many respects, was supplanted by
a kind of Platonism, which was regarded in succeeding centuries as
the “true” Pythagoreanism. Since the content was mostly just passed
along with no other justification than authority, and was accepted
without further examination, it corresponded in its function to the
unified conception of the world characteristic of an carlier, pre-scientific
76 Number symbolism belongs, for Plato, in the realm of myth; and this is true, in
particular, of the famous “nuptial number” of Rep. 545e. (See esp. Heath, Math. 1
3058, A. Diés, Le nombre de Platon [Paris, 1936]; more recently, A. Ahlvers, Zab wid
Klang bei Platon [Bern, 1952] 11ff; M. Denkinger, “L’énigme du nombre de Platon,"’
REG 68 [1955] 38-76; F. von Ehrenfels, AGP 44 [1962] 240-244. Clearchus had already
comniented on the passage, as shown by frr. 3-4 Wehrli.) The Muses present the exposi-
tion of the nuptial number παίζουσαι καὶ ἐρεσχηλοῦσαι, as Plato has it (Rep. §45¢; this is
the merriment of the superior; cf. Phdr. 236b, Phib. 53c, Leg. 885c). What follows is not
nonsense but the indication of meaningful mathematical connections. A relationship
with Pythagoreanism is probable (above, ch. VI 2, n. 13; it is hard to decide whether
Alex. Met. 75.27ff is genuine tradition or a secondary reconstruction). But this does not
mean that whoever has discovered the number 12,960,000, or some other number, will
have grasped the secret of the cosmos, and put himself in a position to prevent its down-
fall. Its decline is not to be arrested by human means (4460). The veil of riddle suggests
an infinite task; it is myth, in the sense of groping for truth at the threshold of the ineff-
able. Even in decline and destruction, it is not chaos or blind conjecture that rules, but an
eternal, intelligent order; and mathematical knowledge can enable us to grasp this
in some small degree. This is the interpretation the Platonist Eratosthenes also gives to
the legend of the “Delian problem,” of duplication of the cube: the god’s intention was,
through the insoluble problem, to inspire interest in the study of geometry (Theo Sim.
2.3ff).
481
LL —<— EEE NG ENE OAN ἡ ἡ σα) τ I FIGBURY
ape. Again, since a doctrine whose appeal is to faith is normally pre-
seated in as ancient garb an possible, the name of Pythagoras sometimes
obscured that of Plato. ‘The tradition of Pythagoras as a philosopher
and scientist is, from the historical point of view, a mistake. But the
fascination that surrounded, and still surrounds, the name of Pythag~
oras docs not come, basically, from specific scientific connotations,
or from the rational method of mathematics, and certainly not from
the success of mathematical physics. More important is the feeling that
there is a kind of knowing which penetrates to the very core of the
universe, which offers truth as something at once beatific and comfort-
ing, and presents the human being as cradled in a universal harmony.
In the figure of Pythagoras an element of Pre-scientific cosmic unity
lives on into an age in which the Greeks were beginning, with their
newly acquired method of rational thought, to make themselves
masters of their world, to call tradition into question, and to abandon
long-cherished beliefs. The price of the new knowledge and freedom
was a loss in inner security; the paths of rational thought lead further
and further in different directions, and into the Boundless. There the
figure of the ancient Sage, who seemed still to possess the secret of
unity, seemed more and more refulgent. Thus after all, there lived on,
in the image of Pythagoras, the great Wizard whom even an advanced
age, though it be unwilling to admit the fact, cannot entirely dismiss.
482
Abbreviations
Bibliography
Indexes
FIGUICVIBEIUT ED
Abbreviations generally follow the standard list of the Oxford Classical
Dictionary (that in the second edition, of 1970, is substantially identical with that
of the first). The following list includes some works frequently cited, some
departures from the usage of OCD, and some items omitted from it. For
Greek authors, see also the list in Liddell-Scott-Jones, Greek-English Lexicon
(Oxford, 1925-40, with Supplement, 1968), and for journals, the list in L’année
philologique.
I. PERIODICALS AND COLLECTIVE WORKS
AA Archdologischer Anzeiger
AA Acta antiqua Academiae Scientiarum Hungaricae (only
cited for works of Szabé; see bibliography)
A&A Antike und Abendland
AAHG Anzeiger fiir die Altertumswissenschaft, hrsg. von der
Oesterreichischen Humanistischen Gesellschaft
ABG Archiv fiir Begriffsgeschichte
AbhBln Abhandlungen der Preussischen Akademie der Wissen-
schaften, Berlin
AbhLpz Abhandlungen der Sachsischen Gesellschaft der Wissen-
schaften, Leipzig
AbhMainz Abhandlungen der geistes- und sozialwissenschaftlichen
Klasse der Akademie der Wissenschaften und der
Literatur, Mainz
AC L’antiquité classique
AGP Archiv fiir Geschichte der Philosophie
AJA American Journal of Archaeology
AJP American Journal of Philology
ALL Archiv fir lateinische Lexikographie und Grammatik
AM Athenische Mitteilungen
ANET Ancient Near Eastern Texts, ed. J. B. Pritchard
AnzAWw Anzeiger fiir die Altertumswissenschaft
485
PERM δ) Κ ἘΠῚ 2 INSING
“δε
nh. dele.
Arch. eph.
ARW
CAG
CAH
CIG
C&M
CMG
CP
CQ
CR
CRAI
DK
Dox.
F’GrHist
FHG
GGN
GRBS
HSCP
HTR
IF
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JAW
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JUS
Kern, Orph. frag.
LS]
ΜΗ
486
Atene ¢ Rama
᾿Δρχαιυλογικὸν δελτίον
᾿Πρχαιυλογικὴ ἐφημερίς
Archiv fur Religionswissenschaft
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page and line)
Cambridee Ancient History
Corpus inscriptionum graccarum
Classica et medievalia
Corpus medicorum graccorum
Classical Philology
Classical Quarterly
Classical Review
Comptes rendues de |’ Académie des Inscriptions et
Belles-lettres
Hermann Diels, Die Fragmente der Vorsokratiker,
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Géttingische gelehrte Nachrichten
Greek, Roman, and Byzantine Studies
Harvard Studies in Classical Philology
Harvard Theological Review
Indogermanische Forschungen
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Jahresbericht iiber die Fortschritte der klassischen
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Otto Kern, Orphicorum fragmenta (Berlin: Weid-
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NED
NGG
NJb
Orph. Frag.
PhilosQ
Phw
PPF
RhGr
RhM
RHR
RivFil
RM
Roscher, Lex.
RPh
SBBln
SBHeid
SBLpz
SBMii
SBWien
SEG
SIG
Studlt
SVF
TAPA
Aboreviations
New Unglish Dictionary (Oxford, tyes)
Nachrichten der Gottinger Gelehrten Ceesellychaft
Neue Jalrbticher fir das klassische Altertum, Geachichte
und deutsche Literatur
sec Kern
Philosophical Quarterly
Philologische Wochenschrift
Hermann Diels, Poetarum philosophorum fragmenta
(Berlin, 1901)
Quellen und Studien zur Geschichte der Mathematik,
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Reallexikon fiir Antike und Christentum
Realencyclopadie der classischen Altertumswissenschaft
Revue des études anciennes
Revue des études grecques
Rhetores graeci, ed. Leonhard [von] Spengel
Rheinisches Museum
Revue de Vhistoire des religions
Rivista di filologia e d’istruzione classica
Rémische Mitteilungen
W. H. Roscher (ed.), Ausfiihrliches Lexikon der
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Sitzungsberichte der Preussischen Akademie der
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Sitzungsberichte der Heidelberger Akademie der
Wissenschaften
Sitzungsberichte der sdchsischen Akademie der Wissen-
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Sitzungsberichte der Bayerischen Akademie der Wissen-
schaften, Munich
Sitzungsberichte der Akademie der Wissenschaften in
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Apul. Plat.
Arist. EE
Arist. EN
Arist. fr.
Arist. Met.
Arist. MM
Aristid. Quint.
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ἐπ ΜΠ} of California’ Publications in Classical
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Wiener Studlen
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Buch der Euklidischen Elemente,’ QSt 3 (1936) $33-553.
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form n.n. (n +1) bei Nikomachos von Gerasa, mit einem Anhang iiber
die platonische Hochzeitszahl,” QSt B 4 (193 8) 181-192.
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